%------------------------------------------------------------------------------
% File     : ITP273_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DeleteBounds 01208_082358
% 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    : 0074_VEBT_DeleteBounds_01208_082358 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 13517 (4356 unt;2596 typ;   0 def)
%            Number of atoms       : 26991 (9772 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 17558 (2584   ~; 553   |;1369   &)
%                                         (2840 <=>;10212  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   23 (   2 avg)
%            Number of FOOLs       : 1786 ( 835 fml; 951 var)
%            Number of X terms     : 1220 (   0  []; 921 ite; 299 let)
%            Number of types       :  540 ( 539 usr)
%            Number of type conns  : 2365 (1589   >; 776   *;   0   +;   0  <<)
%            Number of predicates  :  123 ( 120 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1960 (1960 usr; 491 con; 0-7 aty)
%            Number of variables   : 27112 (26249   !; 863   ?;27112   :)
% 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 14:25:00.092
%------------------------------------------------------------------------------
% Could-be-implicit typings (539)
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    fun_num_real: $tType ).

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

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

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

tff(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
    set_list_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_o: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__fun_It__List__Olist_I_Eo_J_M_Eo_J,type,
    fun_list_o_o: $tType ).

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

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

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

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

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

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

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

tff(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    set_set_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_M_Eo_J,type,
    fun_set_o_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_I_Eo_J_J,type,
    fun_o_set_o: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__fun_I_Eo_Mt__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__Extended____Nat__Oenat,type,
    extended_enat: $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__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

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

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

tff(ty_n_t__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 (2057)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: real > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: set_complex > fun_co51012501629074457plex_o ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: set_real > fun_re6988797352886916895real_o ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_complex > fun_complex_o ).

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

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

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

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_real_real ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_int_int ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_complex_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_int_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_real_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_Pr958786334691620121nt_int > fun_fu3068557634114553409_int_o ).

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

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: vEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_Pr9018117161326028078_int_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_co1148497455491383662plex_o ).

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

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: vEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: product_prod_int_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: nat > fun_se382663106265794036_nat_o ).

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

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: nat > fun_fu4001317426178549763_nat_o ).

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

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_real_real ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_int_int ).

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

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_real > fun_fu4757868175734369219real_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_real > fun_fu8602227297639178947real_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_real > fun_fu4472351043248340291real_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_nat > fun_fu3780232916559739075_nat_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_int > fun_fu2620374121714387907_int_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_complex > fun_fu3074144567237359299plex_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_o_o > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_Pr4796943543331071514_nat_o > fun_fu3843450905092635034_nat_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_o_67843798866261420at_nat > fun_fu4108308376836824364at_nat ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_Pr2118491707810114563at_nat > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_Pr4796943543331071514_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

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

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

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: $o > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_Pr3238374608701176162_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_complex_o > fun_complex_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_set_nat_o > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: set_Pr448751882837621926eger_o > fun_Co480733329897759773un_o_o ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: set_Pr8218934625190621173um_num > fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: set_Pr6200539531224447659at_num > fun_nat_fun_num_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_complex_int > fun_complex_complex ).

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

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_int_fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_co7673397065115134084omplex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

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

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

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

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_nat_real > fun_re4238438383274954274t_real ).

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

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

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: set_int > fun_fu7328815047503086531_int_o ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: set_int > fun_fu8404755328348021045al_int ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: set_real > fun_fu6324116013759063093nt_int ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: set_int > fun_fu5646853623633916995_int_o ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: set_int > fun_fu810243669753863363_int_o ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: set_int > fun_fu8001260946965626035ex_int ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: set_complex > fun_fu2339071201290874307plex_o ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: set_complex > fun_fu7424888833511495217nt_int ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: set_complex > fun_fu3043630359712370243plex_o ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: set_complex > fun_fu7433039812794084273omplex ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: set_real > fun_fu1853463631764337203omplex ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: set_complex > fun_fu5273811730930669507plex_o ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: set_complex > fun_fu6938252901338612493omplex ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: set_nat > fun_fu3656911019502402539omplex ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: set_complex > fun_fu4149201653251398851plex_o ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: set_complex > fun_fu7724386853639975089omplex ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: set_int > fun_fu260611630652282675omplex ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: set_nat > fun_fu5289434066392675523_nat_o ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: set_nat > fun_fu1580565752866428867_nat_o ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: set_nat > fun_fu5253574275612187245l_real ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: set_real > fun_fu5321668254738230545t_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_real_complex > fun_fu3532930458344041007omplex ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_complex > fun_fu4302481349826706223omplex ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_int_complex > fun_fu3558165918884697647omplex ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_complex_rat > fun_fu4396998534562958809ex_rat ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_int_complex > fun_int_o ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_complex_int > fun_complex_o ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_complex > fun_nat_o ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_rat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: vEBT_VEBT > fun_fu4141963935195952137T_real ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: vEBT_VEBT > fun_fu4463953228700081077BT_rat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: vEBT_VEBT > fun_fu5586652545743564717BT_nat ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: vEBT_VEBT > fun_fu7059288016784409353BT_int ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: int > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: int > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_nat_real > fun_na7718645245721260691t_real ).

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

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_rat > fun_nat_rat ).

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

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

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

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

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_nat_nat > fun_nat_Code_integer ).

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

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_complex_complex > fun_complex_real ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_nat_int ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_nat_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_nat_o > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: set_nat > fun_fu40286998686754419omplex ).

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

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_co8378349014452122532t_real > fun_na5723442673449079972x_real ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: set_complex > fun_fu7426956082801719578t_real ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: set_real > fun_fu5104398059618711576t_real ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_int_fun_nat_real > fun_nat_fun_int_real ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: set_int > fun_fu7141299348810748056t_real ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_re2084383175013312036omplex > fun_na1957955181367619364omplex ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: set_real > fun_fu847012557204443036omplex ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_na1142814060743658952omplex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: set_nat > fun_fu7808106308057563584omplex ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_in8056368165505483812omplex > fun_na5558608354090810916omplex ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: set_int > fun_fu6465309499045537820omplex ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_int_fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: set_int > fun_fu688336635129761048at_int ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_co3258106519541737638omplex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: set_complex > fun_fu689363623238371998omplex ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: set_nat > fun_fu7979797992566127108at_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: set_nat > fun_fu2115079063411676732t_real ).

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

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: $o > fun_o_o ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_co8378349014452122532t_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_int_fun_nat_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_re2084383175013312036omplex > fun_real_complex ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_na1142814060743658952omplex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_in8056368165505483812omplex > fun_int_complex ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_int_fun_nat_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_co3258106519541737638omplex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_o_nat > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_nat_Code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_complex_real > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_real_o ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_real_fun_int_o ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: complex > fun_na1554462459066888140plex_o ).

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

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: int > fun_in5570076352459579423_int_o ).

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

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_Co3583066429914935172er_nat ).

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

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_nat_o ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: $o > fun_o_5488095450495718970o_char ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: $o > fun_o_6360995121869435791o_char ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: $o > fun_o_2683029531876921444o_char ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: $o > fun_o_3037882179896959673o_char ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: $o > fun_o_335877185394996366o_char ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: $o > fun_o_2016307092959984611o_char ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: $o > fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_int_nat ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_real_real ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_real_real ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_nat_real ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: fun_nat_real ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_real_real ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: fun_real_o > fun_real_fun_real_o ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option_num ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: fun_num_fun_num_num ).

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun_Pr4799464687642144302_num_o ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1345352211410354436nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: fun_nat_fun_int_int ).

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_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 * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se8260200283734997820nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: fun_int_fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_o ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option_num ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: fun_num_Code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun_num_Code_integer ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Code__Numeral_Opositive,type,
    code_positive: fun_num_Code_integer ).

tff(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: fun_num_int ).

tff(sy_c_Code__Target__Int_Opositive,type,
    code_Target_positive: fun_num_int ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
    comple4887499456419720421f_real: set_real > real ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
    complete_Sup_Sup_int: set_int > int ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

tff(sy_c_Complex_OArg,type,
    arg: fun_complex_real ).

tff(sy_c_Complex_Ocis,type,
    cis: fun_real_complex ).

tff(sy_c_Complex_Ocnj,type,
    cnj: fun_complex_complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > fun_real_complex ).

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: fun_complex_real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: fun_complex_real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: fun_complex_complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
    differ6690327859849518006l_real: ( fun_real_real * filter_real ) > $o ).

tff(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( fun_real_real * real * filter_real ) > $o ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: fun_Pr982841511892225912nt_int ).

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: fun_int_fun_int_int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: fun_na4138050610385705875at_nat ).

tff(sy_c_Divides_Oeucl__rel__int,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_List_Oreplicate_001_Eo,type,
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tff(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
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tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
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tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
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tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Oupto,type,
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tff(sy_c_List_Oupto__aux,type,
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tff(sy_c_List_Oupto__rel,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Onat_Opred,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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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_I_Eo_J,type,
    ord_less_set_o: ( set_o * set_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: ( set_Code_integer * set_Code_integer ) > $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__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: set_set_nat > fun_set_set_nat_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__fun_It__Complex__Ocomplex_M_Eo_J,type,
    ord_le4409593196786536943plex_o: ( fun_complex_o * fun_complex_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Int__Oint_M_Eo_J,type,
    ord_less_fun_int_o: ( fun_int_o * fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Nat__Onat_M_Eo_J,type,
    ord_less_fun_nat_o: ( fun_nat_o * fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Real__Oreal_M_Eo_J,type,
    ord_less_fun_real_o: ( fun_real_o * fun_real_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le921202756137246487_nat_o: ( fun_set_nat_o * fun_set_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
    ord_less_eq_o: $o > fun_o_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__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
    ord_le4104064031414453916r_real: ( filter_real * filter_real ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_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 * set_Code_integer ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: set_complex > fun_set_complex_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > fun_set_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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__Code____Numeral__Ointeger_M_Eo_J_J,type,
    ord_le8980329558974975238eger_o: ( set_Pr448751882837621926eger_o * set_Pr448751882837621926eger_o ) > $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__Nat__Onat_Mt__Num__Onum_J_J,type,
    ord_le8085105155179020875at_num: ( set_Pr6200539531224447659at_num * set_Pr6200539531224447659at_num ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
    ord_le880128212290418581um_num: ( set_Pr8218934625190621173um_num * set_Pr8218934625190621173um_num ) > $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__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > fun_set_set_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_M_Eo_J_J,type,
    ord_le2909477399203599485un_o_o: ( fun_Co480733329897759773un_o_o * fun_Co480733329897759773un_o_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    ord_le7644322562001619171plex_o: ( fun_complex_o * fun_complex_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_M_Eo_J,type,
    ord_le5418975059635765733_int_o: ( fun_int_o * fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    ord_le77980682512674322_int_o: ( fun_int_fun_int_o * fun_int_fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__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__Nat__Onat_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    ord_le1526520017675560704_num_o: ( fun_nat_fun_num_o * fun_nat_fun_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    ord_le3306795289226699318_num_o: ( fun_num_fun_num_o * fun_num_fun_num_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__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le6468474429079055883_nat_o: ( fun_set_nat_o * fun_set_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
    ord_max_Code_integer: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: fun_Ex8341929632020906428d_enat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: fun_num_fun_num_num ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: rat > fun_rat_rat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: real > fun_real_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: ( set_int * set_int ) > set_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: ( set_nat * set_nat ) > set_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_max_set_real: ( set_real * set_real ) > set_real ).

tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: fun_nat_o > nat ).

tff(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_9091379641038594480t_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: fun_nat_nat > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_mono_nat_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_mono_real_real: fun_real_real > $o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

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_Product__Type_OPair_001_Eo_001_Eo,type,
    product_Pair_o_o: ( $o * $o ) > product_prod_o_o ).

tff(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
    product_Pair_o_int: ( $o * int ) > product_prod_o_int ).

tff(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    produc2982872950893828659T_VEBT: ( $o * vEBT_VEBT ) > produc2504756804600209347T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc6677183202524767010eger_o: ( code_integer * $o ) > produc6271795597528267376eger_o ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: fun_in1696471567231710611nt_int ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: fun_na4138050610385705875at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
    product_Pair_nat_num: ( nat * num ) > product_prod_nat_num ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: ( nat * product_prod_nat_nat ) > produc7248412053542808358at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    produc1195630363706982562at_num: ( nat * product_prod_nat_num ) > produc2963631642982155120at_num ).

tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
    produc5098337634421038937on_nat: ( option_nat * option_nat ) > produc4953844613479565601on_nat ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
    produc8585076106096196333on_num: ( option_num * option_num ) > produc3447558737645232053on_num ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc488173922507101015at_nat: ( option4927543243414619207at_nat * option4927543243414619207at_nat ) > produc6121120109295599847at_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc8721562602347293563VEBT_o: ( vEBT_VEBT * $o ) > produc334124729049499915VEBT_o ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: ( vEBT_VEBT * int ) > produc4894624898956917775BT_int ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: ( vEBT_VEBT * nat ) > produc9072475918466114483BT_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
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tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_Oinsert_001t__Rat__Orat,type,
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tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001_Eo,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
    set_or5849166863359141190n_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
    set_ord_lessThan_o: fun_o_set_o ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: fun_int_set_int ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
    set_ord_lessThan_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or890127255671739683et_nat: set_nat > set_set_nat ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o ) > fun_o_char ).

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
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tff(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: ( filter_real * fun_real_real ) > $o ).

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    topolo5044208981011980120l_real: ( set_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Ogenerate__topology_001_Eo,type,
    topolo4667128019001906403logy_o: set_set_o > fun_set_o_o ).

tff(sy_c_Topological__Spaces_Ogenerate__topology_001t__Int__Oint,type,
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tff(sy_c_Topological__Spaces_Ogenerate__topology_001t__Nat__Onat,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001_Eo,type,
    topolo9180104560040979295open_o: fun_set_o_o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
    topolo4110288021797289639omplex: fun_set_complex_o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Int__Oint,type,
    topolo4325760605701065253en_int: fun_set_int_o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Nat__Onat,type,
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tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Real__Oreal,type,
    topolo4860482606490270245n_real: fun_set_real_o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
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tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_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: real > fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > fun_real_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_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__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__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: vEBT_VEBT > fun_nat_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__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__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_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 * option_num ) > option_num ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( fun_Pr2477239776880934807at_nat * option4927543243414619207at_nat * option4927543243414619207at_nat ) > option4927543243414619207at_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: fun_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_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V7235779383477046023at_nat: fun_Pr4779616527126863150_nat_o ).

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 * nat ) > option_nat ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_nat_o ).

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 * 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,
    accp_P1096762738010456898nt_int: fun_Pr9018117161326028078_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: fun_Pr2028404973800837934_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: fun_Pr4799464687642144302_num_o > fun_Pr6331313853612757382_num_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: fun_Pr8877776288780692068_nat_o > fun_Pr7546709192626087934_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P4694664897847628410at_nat: ( fun_Pr8463018485500633518_nat_o * produc9217583074051984497at_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
    accp_P5526603486790338531on_nat: ( fun_Pr1283576600272279410_nat_o * produc7004362771917097900on_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
    accp_P3591886921626383704at_num: ( fun_Pr7588637180273442094_num_o * produc8114805097830739791at_num ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    accp_P1225658569124232473on_num: ( fun_Pr4822914654802553862_num_o * produc2703417854250991842on_num ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    accp_P4760198183577772864at_nat: ( fun_Pr4779616527126863150_nat_o * produc1260151622200656695at_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: fun_VE2962686792283380846VEBT_o > fun_VEBT_VEBT_o ).

tff(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
    measure_int: fun_int_nat > set_Pr958786334691620121nt_int ).

tff(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: fun_nat_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Omeasure_001t__Num__Onum,type,
    measure_num: fun_num_nat > set_Pr8218934625190621173um_num ).

tff(sy_c_aa_001_Eo_001_Eo,type,
    aa_o_o: ( fun_o_o * $o ) > $o ).

tff(sy_c_aa_001_Eo_001t__Int__Oint,type,
    aa_o_int: ( fun_o_int * $o ) > int ).

tff(sy_c_aa_001_Eo_001t__Nat__Onat,type,
    aa_o_nat: ( fun_o_nat * $o ) > nat ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_o_P3134438092330336605nt_int: ( fun_o_8988054265482332260nt_int * $o ) > product_prod_int_int ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_o_P6312935399938478757at_nat: ( fun_o_67843798866261420at_nat * $o ) > product_prod_nat_nat ).

tff(sy_c_aa_001_Eo_001t__Set__Oset_I_Eo_J,type,
    aa_o_set_o: ( fun_o_set_o * $o ) > set_o ).

tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    aa_o_set_complex: ( fun_o_set_complex * $o ) > set_complex ).

tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
    aa_o_set_int: ( fun_o_set_int * $o ) > set_int ).

tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
    aa_o_set_nat: ( fun_o_set_nat * $o ) > set_nat ).

tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
    aa_o_set_real: ( fun_o_set_real * $o ) > set_real ).

tff(sy_c_aa_001_Eo_001t__String__Ochar,type,
    aa_o_char: ( fun_o_char * $o ) > char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J_J_J_J_J_J,type,
    aa_o_f3038258172971468011o_char: ( fun_o_5488095450495718970o_char * $o ) > fun_o_6360995121869435791o_char ).

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    aa_o_f1143307120959153672o_char: ( fun_o_6360995121869435791o_char * $o ) > fun_o_2683029531876921444o_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J_J_J_J,type,
    aa_o_f2932186792858379797o_char: ( fun_o_2683029531876921444o_char * $o ) > fun_o_3037882179896959673o_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J_J_J,type,
    aa_o_f4183413333637126194o_char: ( fun_o_3037882179896959673o_char * $o ) > fun_o_335877185394996366o_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J_J,type,
    aa_o_f9003794812476387647o_char: ( fun_o_335877185394996366o_char * $o ) > fun_o_2016307092959984611o_char ).

tff(sy_c_aa_001_Eo_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J,type,
    aa_o_f2742674633914528860o_char: ( fun_o_2016307092959984611o_char * $o ) > fun_Co681828205884031672o_char ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001_Eo,type,
    aa_Code_integer_o: ( fun_Code_integer_o * code_integer ) > $o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    aa_Cod1783974652533396728nteger: ( fun_Co8499238673658658103nteger * code_integer ) > code_integer ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    aa_Code_integer_int: ( fun_Code_integer_int * code_integer ) > int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    aa_Code_integer_nat: ( fun_Code_integer_nat * code_integer ) > nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    aa_Code_integer_num: ( fun_Code_integer_num * code_integer ) > num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_Cod3168588076859124107eger_o: ( fun_Co1766037875893669450eger_o * code_integer ) > produc6271795597528267376eger_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod6977894654539884864nteger: ( fun_Co6597186368970283863nteger * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_I_Eo_M_Eo_J,type,
    aa_Cod6379977994291289310un_o_o: ( fun_Co480733329897759773un_o_o * code_integer ) > fun_o_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_I_Eo_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
    aa_Cod2779864889637189172omplex: ( fun_Co5015990272074005067omplex * code_integer ) > fun_o_set_complex ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_I_Eo_Mt__Set__Oset_It__Int__Oint_J_J,type,
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    aa_fun7274880274332377722t_real: ( fun_fu5321668254738230545t_real * fun_re4238438383274954274t_real ) > fun_fu8372050067615010778t_real ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    aa_fun6240130701872165846l_real: ( fun_fu5253574275612187245l_real * fun_re4238438383274954274t_real ) > fun_fu4971871526818099510l_real ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    aa_fun751667557118977660_nat_o: ( fun_fu2408719596384163971_nat_o * fun_set_nat_o ) > fun_set_nat_o ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    aa_fun3386010016858254564at_nat: ( fun_fu1590586450949517859at_nat * fun_set_nat_nat ) > fun_set_nat_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    aa_fun8193769468198689164VEBT_o: ( fun_fu6329627000648617163VEBT_o * fun_VEBT_VEBT_o ) > fun_VEBT_VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    aa_fun7144222995531728252BT_int: ( fun_fu5786399312697532803BT_int * fun_VEBT_VEBT_int ) > fun_VEBT_VEBT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    aa_fun7485955581636121530BT_int: ( fun_fu7059288016784409353BT_int * fun_VEBT_VEBT_int ) > fun_fu5786399312697532803BT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    aa_fun8284231930576791164BT_nat: ( fun_fu5238640373165125763BT_nat * fun_VEBT_VEBT_nat ) > fun_VEBT_VEBT_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    aa_fun8778301808485728350BT_nat: ( fun_fu5586652545743564717BT_nat * fun_VEBT_VEBT_nat ) > fun_fu5238640373165125763BT_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J,type,
    aa_fun8105832842879916156BT_rat: ( fun_fu2188483506383969923BT_rat * fun_VEBT_VEBT_rat ) > fun_VEBT_VEBT_rat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_J,type,
    aa_fun2693410532875676774BT_rat: ( fun_fu4463953228700081077BT_rat * fun_VEBT_VEBT_rat ) > fun_fu2188483506383969923BT_rat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    aa_fun1825138017709814396T_real: ( fun_fu8198533418994508931T_real * fun_VEBT_VEBT_real ) > fun_VEBT_VEBT_real ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
    aa_fun910392650362481850T_real: ( fun_fu4141963935195952137T_real * fun_VEBT_VEBT_real ) > fun_fu8198533418994508931T_real ).

tff(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_o > real ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: fun_co1148497455491383662plex_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__Set__Oset_It__Nat__Onat_J,type,
    fequal_set_nat: set_nat > fun_set_nat_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__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__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $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__Code____Numeral__Ointeger_M_Eo_J,type,
    member1379723562493234055eger_o: ( produc6271795597528267376eger_o * set_Pr448751882837621926eger_o ) > $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_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__Num__Onum_J,type,
    member9148766508732265716at_num: ( product_prod_nat_num * set_Pr6200539531224447659at_num ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    member7279096912039735102um_num: ( product_prod_num_num * set_Pr8218934625190621173um_num ) > $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__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > fun_set_set_nat_o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: vEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_v_a____,type,
    a: nat ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

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_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10905)
tff(fact_0_True,axiom,
    xa = ma ).

% True
tff(fact_1_False,axiom,
    xa != mi ).

% False
tff(fact_2__C5_Ohyps_C_I7_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% "5.hyps"(7)
tff(fact_3_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_4_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_5_deggy,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg) ).

% deggy
tff(fact_6__092_060open_062mi_A_092_060le_062_Ax_A_092_060and_062_Ax_A_092_060le_062_Ama_092_060close_062,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),xa)
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,xa),ma) ) ).

% \<open>mi \<le> x \<and> x \<le> ma\<close>
tff(fact_7_tdeletemimi,axiom,
    ! [Dega: 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))),Dega)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Mia)),Dega,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_8_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_9__C7_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ).

% "7"
tff(fact_10__092_060open_062T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_A_ItreeList_A_091high_Ax_A_Ideg_Adiv_A2_J_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_093_A_B_Athe_A_Ivebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_A_092_060le_062_A3_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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(vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% \<open>T\<^sub>m\<^sub>a\<^sub>x\<^sub>t (treeList [high x (deg div 2) := vebt_delete (treeList ! high x (deg div 2)) (low x (deg div 2))] ! the (vebt_maxt (vebt_delete summary (high x (deg div 2))))) \<le> 3\<close>
tff(fact_11__C12_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa)),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_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,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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(vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ).

% "12"
tff(fact_12_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_13_add__self__div__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Mb ).

% add_self_div_2
tff(fact_14_nat__add__left__cancel__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),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,Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_15_semiring__norm_I86_J,axiom,
    ! [Mb: num] : aa_num_num(bit1,Mb) != one ).

% semiring_norm(86)
tff(fact_16_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_17_semiring__norm_I89_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit1,Mb) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_18_semiring__norm_I88_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit0,Mb) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_19_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_20_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_21_field__sum__of__halves,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_22_field__sum__of__halves,axiom,
    ! [Xb: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_23_semiring__norm_I85_J,axiom,
    ! [Mb: num] : aa_num_num(bit0,Mb) != one ).

% semiring_norm(85)
tff(fact_24_semiring__norm_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(6)
tff(fact_25_semiring__norm_I87_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit0,Mb) = aa_num_num(bit0,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(87)
tff(fact_26_semiring__norm_I90_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit1,Mb) = aa_num_num(bit1,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(90)
tff(fact_27_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_28_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_29_semiring__norm_I7_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(7)
tff(fact_30_semiring__norm_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(9)
tff(fact_31_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_32_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_33_semiring__norm_I5_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),one) = aa_num_num(bit1,Mb) ).

% semiring_norm(5)
tff(fact_34_semiring__norm_I8_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one)) ).

% semiring_norm(8)
tff(fact_35_semiring__norm_I10_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),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,Mb),Nb)),one)) ).

% semiring_norm(10)
tff(fact_36_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => ? [X2: nat] :
            ( aa_nat_o(P,X2)
            & ! [Y2: nat] :
                ( aa_nat_o(P,Y2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_37_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% nat_le_linear
tff(fact_38_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_39_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% eq_imp_le
tff(fact_40_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_41_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_42_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ? [K2: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2) ) ).

% nat_le_iff_add
tff(fact_43_trans__le__add2,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_le_add2
tff(fact_44_trans__le__add1,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_le_add1
tff(fact_45_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_46_mem__Collect__eq,axiom,
    ! [Aa: real,P: fun_real_o] :
      ( aa_set_real_o(member_real(Aa),collect_real(P))
    <=> aa_real_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_47_mem__Collect__eq,axiom,
    ! [Aa: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(Aa,collec213857154873943460nt_int(P))
    <=> aa_Pro4592068669816069595_int_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_48_mem__Collect__eq,axiom,
    ! [Aa: complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(Aa),collect_complex(P))
    <=> aa_complex_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_49_mem__Collect__eq,axiom,
    ! [Aa: set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(Aa),collect_set_nat(P))
    <=> aa_set_nat_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_50_mem__Collect__eq,axiom,
    ! [Aa: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Aa),collect_nat(P))
    <=> aa_nat_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_51_mem__Collect__eq,axiom,
    ! [Aa: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(Aa),collect_int(P))
    <=> aa_int_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_52_Collect__mem__eq,axiom,
    ! [A: set_real] : collect_real(aTP_Lamm_a(A)) = A ).

% Collect_mem_eq
tff(fact_53_Collect__mem__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_aa(A)) = A ).

% Collect_mem_eq
tff(fact_54_Collect__mem__eq,axiom,
    ! [A: set_complex] : collect_complex(aTP_Lamm_ab(A)) = A ).

% Collect_mem_eq
tff(fact_55_Collect__mem__eq,axiom,
    ! [A: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A)) = A ).

% Collect_mem_eq
tff(fact_56_Collect__mem__eq,axiom,
    ! [A: set_nat] : collect_nat(aTP_Lamm_ad(A)) = A ).

% Collect_mem_eq
tff(fact_57_Collect__mem__eq,axiom,
    ! [A: set_int] : collect_int(aTP_Lamm_ae(A)) = A ).

% Collect_mem_eq
tff(fact_58_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_59_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_60_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X2: set_nat] :
          ( aa_set_nat_o(P,X2)
        <=> aa_set_nat_o(Q,X2) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_61_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_62_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_63_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_64_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).

% le_Suc_ex
tff(fact_65_add__leD2,axiom,
    ! [Mb: 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,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% add_leD2
tff(fact_66_add__leD1,axiom,
    ! [Mb: 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,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% add_leD1
tff(fact_67_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% le_add2
tff(fact_68_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) ).

% le_add1
tff(fact_69_add__leE,axiom,
    ! [Mb: 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,Mb),K)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% add_leE
tff(fact_70_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_71_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K)) ) ).

% div_le_mono
tff(fact_72_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_73_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_74_Some,axiom,
    vEBT_vebt_maxt(vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) = some_nat(a) ).

% Some
tff(fact_75_one__add__one,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_76_one__add__one,axiom,
    aa_real_real(plus_plus_real(one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_77_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_78_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_79_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_80_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_81_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_82_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_83_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_84_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_85_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_86_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_87_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_88_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_89_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_90_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_91_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_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(aa_num_fun_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(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_94_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_95_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_96_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_97_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_98_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_99_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_100_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_101_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_102_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_103_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_104_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_105_delt__out__of__range,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: 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))),Dega)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% delt_out_of_range
tff(fact_106_del__single__cont,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya) ) ) ) ).

% del_single_cont
tff(fact_107_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_108_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_109_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_110_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_111_delete__pres__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(Ta,Xb)),Nb) ) ).

% delete_pres_valid
tff(fact_112_dele__bmo__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_delete(Ta,Xb)),Y3)
      <=> ( ( Xb != Y3 )
          & aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Y3) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_113_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
     => aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb) ) ).

% maxbmo
tff(fact_114__C5_Ohyps_C_I1_J,axiom,
    aa_nat_o(vEBT_invar_vebt(summary),m) ).

% "5.hyps"(1)
tff(fact_115_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( numera6690914467698888265omplex(Mb) = numera6690914467698888265omplex(Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_116_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Mb) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_117_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Mb) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_118_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Mb) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_119_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Mb) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_120_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT,Y3: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I,Y3) = list_u1324408373059187874T_VEBT(Xs,I,Y3) ).

% list_update_overwrite
tff(fact_121_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_122_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_123_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_124_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_125_nat__add__left__cancel__less,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_126_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_127_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_128_semiring__norm_I71_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(71)
tff(fact_129_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).

% semiring_norm(68)
tff(fact_130_semiring__norm_I73_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(73)
tff(fact_131_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_132_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_133_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_134_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_135_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_136_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_137_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_138_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_139_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_140_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_141_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_142_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_143_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_144_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_145_semiring__norm_I69_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),one) ).

% semiring_norm(69)
tff(fact_146_semiring__norm_I72_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(72)
tff(fact_147_semiring__norm_I70_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),one) ).

% semiring_norm(70)
tff(fact_148_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_149_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_150_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(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_151_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(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_152_lesseq__shift,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,some_nat(Xb)),some_nat(Y3)) ) ).

% lesseq_shift
tff(fact_153__C5_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X3: vEBT_VEBT] :
        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(treeList))
       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) ) ).

% "5.hyps"(6)
tff(fact_154__C5_Ohyps_C_I8_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "5.hyps"(8)
tff(fact_155__092_060open_062high_Ax_A_Ideg_Adiv_A2_J_A_060_Alength_AtreeList_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList)) ).

% \<open>high x (deg div 2) < length treeList\<close>
tff(fact_156_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_157_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_158_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_159_less__not__refl3,axiom,
    ! [S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),Ta)
     => ( S != Ta ) ) ).

% less_not_refl3
tff(fact_160_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_161_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
             => aa_nat_o(P,M) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_162_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
              & ~ aa_nat_o(P,M) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_163_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ( Xb != Y3 )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_164_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_165_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_166_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_167_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_168_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),one)
    <=> ( Xb = one ) ) ).

% le_num_One_iff
tff(fact_169_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_170_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_171_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_172_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_173_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_174_trans__less__add1,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_less_add1
tff(fact_175_trans__less__add2,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_less_add2
tff(fact_176_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Mb: 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,Mb),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,Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_177_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_178_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_179_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_180_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_181_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ( Mb != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_182_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_183_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_184_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_185_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_186_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_187_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Mb: nat,K: nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M2)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,Mb)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))) ) ).

% mono_nat_linear_lb
tff(fact_188_is__num__normalize_I1_J,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(Aa),B)),C) = aa_real_real(plus_plus_real(Aa),aa_real_real(plus_plus_real(B),C)) ).

% is_num_normalize(1)
tff(fact_189_is__num__normalize_I1_J,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Aa),B)),C) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(B),C)) ).

% is_num_normalize(1)
tff(fact_190_is__num__normalize_I1_J,axiom,
    ! [Aa: 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),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% is_num_normalize(1)
tff(fact_191_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_192_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,
    ! [Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Uu) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_193_list__update__swap,axiom,
    ! [I: nat,I3: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,X4: vEBT_VEBT] :
      ( ( I != I3 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I3,X4) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I3,X4),I,Xb) ) ) ).

% list_update_swap
tff(fact_194_field__less__half__sum,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Y3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_195_field__less__half__sum,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Y3)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_196_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_197_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_198_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_199_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_200_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) ).

% one_le_numeral
tff(fact_201_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_le_numeral
tff(fact_202_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_203_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_204_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_205_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_Bit0
tff(fact_206_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_Bit0
tff(fact_207_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_208_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_209_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),numera6690914467698888265omplex(Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Xb)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_210_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_211_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_212_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_213_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_214_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_215_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_216_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_217_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_218_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_219_divide__numeral__1,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),numera6690914467698888265omplex(one)) = Aa ).

% divide_numeral_1
tff(fact_220_divide__numeral__1,axiom,
    ! [Aa: real] : aa_real_real(divide_divide_real(Aa),aa_num_real(numeral_numeral_real,one)) = Aa ).

% divide_numeral_1
tff(fact_221_divide__numeral__1,axiom,
    ! [Aa: rat] : aa_rat_rat(divide_divide_rat(Aa),aa_num_rat(numeral_numeral_rat,one)) = Aa ).

% divide_numeral_1
tff(fact_222_num_Oexhaust,axiom,
    ! [Y3: num] :
      ( ( Y3 != one )
     => ( ! [X22: num] : Y3 != aa_num_num(bit0,X22)
       => ~ ! [X32: num] : Y3 != aa_num_num(bit1,X32) ) ) ).

% num.exhaust
tff(fact_223_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_224_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_225_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_226_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_227_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_228_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_229_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_230_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xb)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) ) ).

% in_children_def
tff(fact_231__092_060open_062invar__vebt_A_Ivebt__delete_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_J_An_092_060close_062,axiom,
    aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),na) ).

% \<open>invar_vebt (vebt_delete (treeList ! high x (deg div 2)) (low x (deg div 2))) n\<close>
tff(fact_232__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
    aa_nat_o(vEBT_invar_vebt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),na) ).

% \<open>invar_vebt (treeList ! high x (deg div 2)) n\<close>
tff(fact_233_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_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) = some_nat(Xb) ) ) ) ).

% maxt_sound
tff(fact_234_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% maxt_corr
tff(fact_235_pred__max,axiom,
    ! [Dega: 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))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = some_nat(Maa) ) ) ) ).

% pred_max
tff(fact_236_succ__min,axiom,
    ! [Dega: 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))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = some_nat(Mia) ) ) ) ).

% succ_min
tff(fact_237_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_238_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_239_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_240_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_241_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_242_minNull__delete__time__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_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(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_243_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_244_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_12) ) ).

% not_min_Null_member
tff(fact_245__C5_Ohyps_C_I4_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "5.hyps"(4)
tff(fact_246_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),set_complex2(Xs))
         => aa_complex_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).

% inthall
tff(fact_247_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_248_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Nb: nat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),set_set_nat2(Xs))
         => aa_set_nat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,nth_set_nat(Xs,Nb)) ) ) ).

% inthall
tff(fact_249_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_250_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_251_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_252_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_253_power__shift,axiom,
    ! [Xb: nat,Y3: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Y3) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xb)),some_nat(Y3)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_254__C5_Ohyps_C_I2_J,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))),m) ).

% "5.hyps"(2)
tff(fact_255_option_Oinject,axiom,
    ! [X23: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X23) = some_P7363390416028606310at_nat(Y22) )
    <=> ( X23 = Y22 ) ) ).

% option.inject
tff(fact_256_option_Oinject,axiom,
    ! [X23: nat,Y22: nat] :
      ( ( some_nat(X23) = some_nat(Y22) )
    <=> ( X23 = Y22 ) ) ).

% option.inject
tff(fact_257_option_Oinject,axiom,
    ! [X23: num,Y22: num] :
      ( ( some_num(X23) = some_num(Y22) )
    <=> ( X23 = Y22 ) ) ).

% option.inject
tff(fact_258_pow__sum,axiom,
    ! [Aa: 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,Aa),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)) = 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_259_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_260_geqmaxNone,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Maa),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_261_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia = Maa )
       => ( ! [X3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) )
          & ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_262__C5_Ohyps_C_I5_J,axiom,
    ! [I4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),X_13)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(summary),I4) ) ) ).

% "5.hyps"(5)
tff(fact_263_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).

% high_bound_aux
tff(fact_264_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = some_nat(Y3) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),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_265_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = some_nat(Y3) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),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_266_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y3)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_267_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_268_not__Some__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ! [Y4: product_prod_nat_nat] : Xb != some_P7363390416028606310at_nat(Y4)
    <=> ( Xb = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_269_not__Some__eq,axiom,
    ! [Xb: option_nat] :
      ( ! [Y4: nat] : Xb != some_nat(Y4)
    <=> ( Xb = none_nat ) ) ).

% not_Some_eq
tff(fact_270_not__Some__eq,axiom,
    ! [Xb: option_num] :
      ( ! [Y4: num] : Xb != some_num(Y4)
    <=> ( Xb = none_num ) ) ).

% not_Some_eq
tff(fact_271_not__None__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ( Xb != none_P5556105721700978146at_nat )
    <=> ? [Y4: product_prod_nat_nat] : Xb = some_P7363390416028606310at_nat(Y4) ) ).

% not_None_eq
tff(fact_272_not__None__eq,axiom,
    ! [Xb: option_nat] :
      ( ( Xb != none_nat )
    <=> ? [Y4: nat] : Xb = some_nat(Y4) ) ).

% not_None_eq
tff(fact_273_not__None__eq,axiom,
    ! [Xb: option_num] :
      ( ( Xb != none_num )
    <=> ? [Y4: num] : Xb = some_num(Y4) ) ).

% not_None_eq
tff(fact_274_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_275_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_276_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_277_semiring__norm_I78_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(78)
tff(fact_278_semiring__norm_I75_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),one) ).

% semiring_norm(75)
tff(fact_279_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Mb: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_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))),Mb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_280_semiring__norm_I80_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(80)
tff(fact_281__C5_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I4 )
           => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
          & ! [X3: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),na) = I4 )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),na)) )
             => ( 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) ) ) ) ) ) ).

% "5.hyps"(9)
tff(fact_282_succ__list__to__short,axiom,
    ! [Dega: 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))),Dega)
     => ( 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_283_pred__list__to__short,axiom,
    ! [Dega: 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))),Dega)
     => ( 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_284_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_285_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_286_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_287_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_288_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Nb)) ).

% semiring_norm(76)
tff(fact_289_semiring__norm_I81_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(81)
tff(fact_290_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_291__C5_Ohyps_C_I3_J,axiom,
    m = aa_nat_nat(suc,na) ).

% "5.hyps"(3)
tff(fact_292_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_293_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_294_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_295_semiring__norm_I74_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(74)
tff(fact_296_semiring__norm_I79_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(79)
tff(fact_297_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_298_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_299_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_300_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Px: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = 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_301_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = 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_302_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_303_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_304_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_305_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_306_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_307_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_308_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),B2)
    <=> ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),set_complex2(Xs))
         => aa_set_complex_o(member_complex(X),B2) ) ) ).

% subset_code(1)
tff(fact_309_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B2)
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_set_real_o(member_real(X),B2) ) ) ).

% subset_code(1)
tff(fact_310_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B2)
    <=> ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),set_set_nat2(Xs))
         => aa_set_set_nat_o(member_set_nat(X),B2) ) ) ).

% subset_code(1)
tff(fact_311_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B2)
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X),B2) ) ) ).

% subset_code(1)
tff(fact_312_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),B2)
    <=> ! [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),B2) ) ) ).

% subset_code(1)
tff(fact_313_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B2)
    <=> ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_set_int_o(member_int(X),B2) ) ) ).

% subset_code(1)
tff(fact_314_size__neq__size__imp__neq,axiom,
    ! [Xb: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xb) != size_s6755466524823107622T_VEBT(Y3) )
     => ( Xb != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_315_size__neq__size__imp__neq,axiom,
    ! [Xb: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Xb) != aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Y3) )
     => ( Xb != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_316_size__neq__size__imp__neq,axiom,
    ! [Xb: num,Y3: num] :
      ( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Y3) )
     => ( Xb != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_317_size__neq__size__imp__neq,axiom,
    ! [Xb: list_o,Y3: list_o] :
      ( ( size_size_list_o(Xb) != size_size_list_o(Y3) )
     => ( Xb != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_318_size__neq__size__imp__neq,axiom,
    ! [Xb: list_int,Y3: list_int] :
      ( ( size_size_list_int(Xb) != size_size_list_int(Y3) )
     => ( Xb != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_319_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) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_320_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)) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_321_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) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_322_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Xb: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I2)) )
     => ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
       => aa_complex_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_323_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xb: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I2)) )
     => ( aa_set_real_o(member_real(Xb),set_real2(Xs))
       => aa_real_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_324_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Xb: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P,nth_set_nat(Xs,I2)) )
     => ( aa_set_set_nat_o(member_set_nat(Xb),set_set_nat2(Xs))
       => aa_set_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_325_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xb: nat] :
      ( ! [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)) )
     => ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
       => aa_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_326_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( ! [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)) )
     => ( 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_327_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xb: $o] :
      ( ! [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)) )
     => ( member_o((Xb),set_o2(Xs))
       => aa_o_o(P,(Xb)) ) ) ).

% all_nth_imp_all_set
tff(fact_328_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xb: int] :
      ( ! [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)) )
     => ( aa_set_int_o(member_int(Xb),set_int2(Xs))
       => aa_int_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_329_in__set__conv__nth,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_330_in__set__conv__nth,axiom,
    ! [Xb: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xb),set_real2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_331_in__set__conv__nth,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xb),set_set_nat2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s3254054031482475050et_nat(Xs))
          & ( nth_set_nat(Xs,I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_332_in__set__conv__nth,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_333_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))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_334_in__set__conv__nth,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I5)
          <=> (Xb) ) ) ) ).

% in_set_conv_nth
tff(fact_335_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xb),set_int2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_336_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_337_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_338_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_339_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(aa_nat_complex(nth_complex(Xs),Nb)),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_340_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_341_nth__mem,axiom,
    ! [Nb: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(nth_set_nat(Xs,Nb)),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_342_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_343_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_344_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_345_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_346_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_complex,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(Xb),set_complex2(list_update_complex(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_347_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_348_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_set_nat,Xb: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(Xb),set_set_nat2(list_update_set_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_349_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_350_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_351_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_352_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_353_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys2) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_354_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_355_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_356_set__update__subsetI,axiom,
    ! [Xs: list_complex,A: set_complex,Xb: complex,I: nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),A)
     => ( aa_set_complex_o(member_complex(Xb),A)
       => aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,Xb))),A) ) ) ).

% set_update_subsetI
tff(fact_357_set__update__subsetI,axiom,
    ! [Xs: list_real,A: set_real,Xb: real,I: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A)
     => ( aa_set_real_o(member_real(Xb),A)
       => aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xb))),A) ) ) ).

% set_update_subsetI
tff(fact_358_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A: set_set_nat,Xb: set_nat,I: nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A)
     => ( aa_set_set_nat_o(member_set_nat(Xb),A)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,Xb))),A) ) ) ).

% set_update_subsetI
tff(fact_359_set__update__subsetI,axiom,
    ! [Xs: list_nat,A: set_nat,Xb: nat,I: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A)
     => ( aa_set_nat_o(member_nat(Xb),A)
       => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xb))),A) ) ) ).

% set_update_subsetI
tff(fact_360_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A: set_VEBT_VEBT,Xb: vEBT_VEBT,I: nat] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),A)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A)
       => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xb))),A) ) ) ).

% set_update_subsetI
tff(fact_361_set__update__subsetI,axiom,
    ! [Xs: list_int,A: set_int,Xb: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A)
     => ( aa_set_int_o(member_int(Xb),A)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xb))),A) ) ) ).

% set_update_subsetI
tff(fact_362_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I5) = nth_VEBT_VEBT(Ys,I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_363_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I5)
            <=> aa_nat_o(nth_o(Ys),I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_364_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I5) = aa_nat_int(nth_int(Ys),I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_365_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_13: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I5),X_13) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I5),nth_VEBT_VEBT(Xs3,I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_366_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_13: $o] : aa_o_o(aa_nat_fun_o_o(P,I5),(X_13)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_o_o(aa_nat_fun_o_o(P,I5),aa_nat_o(nth_o(Xs3),I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_367_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_13: int] : aa_int_o(aa_nat_fun_int_o(P,I5),X_13) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_int_o(aa_nat_fun_int_o(P,I5),aa_nat_int(nth_int(Xs3),I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_368_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( 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) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_369_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( 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) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_370_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( 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) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_371_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_372_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_373_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_374_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_375_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_376_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_377_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y3: option4927543243414619207at_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y3) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y3) )
       => ( ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A2) )
             => ( ( Y3 = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y3) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y3) ) ) ) ).

% combine_options_cases
tff(fact_378_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y3: option_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y3) )
     => ( ( ( Y3 = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y3) )
       => ( ! [A2: product_prod_nat_nat,B3: nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A2) )
             => ( ( Y3 = some_nat(B3) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y3) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y3) ) ) ) ).

% combine_options_cases
tff(fact_379_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y3: option_num] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y3) )
     => ( ( ( Y3 = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y3) )
       => ( ! [A2: product_prod_nat_nat,B3: num] :
              ( ( Xb = some_P7363390416028606310at_nat(A2) )
             => ( ( Y3 = some_num(B3) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y3) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y3) ) ) ) ).

% combine_options_cases
tff(fact_380_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op3832135881594366065_nat_o,Y3: option4927543243414619207at_nat] :
      ( ( ( Xb = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y3) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y3) )
       => ( ! [A2: nat,B3: product_prod_nat_nat] :
              ( ( Xb = some_nat(A2) )
             => ( ( Y3 = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y3) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y3) ) ) ) ).

% combine_options_cases
tff(fact_381_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op5794009454290973322_nat_o,Y3: option_nat] :
      ( ( ( Xb = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y3) )
     => ( ( ( Y3 = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y3) )
       => ( ! [A2: nat,B3: nat] :
              ( ( Xb = some_nat(A2) )
             => ( ( Y3 = some_nat(B3) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y3) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y3) ) ) ) ).

% combine_options_cases
tff(fact_382_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op6120599277751319488_num_o,Y3: option_num] :
      ( ( ( Xb = none_nat )
       => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y3) )
     => ( ( ( Y3 = none_num )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y3) )
       => ( ! [A2: nat,B3: num] :
              ( ( Xb = some_nat(A2) )
             => ( ( Y3 = some_num(B3) )
               => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y3) ) )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y3) ) ) ) ).

% combine_options_cases
tff(fact_383_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op5373868528577402299_nat_o,Y3: option4927543243414619207at_nat] :
      ( ( ( Xb = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y3) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y3) )
       => ( ! [A2: num,B3: product_prod_nat_nat] :
              ( ( Xb = some_num(A2) )
             => ( ( Y3 = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y3) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y3) ) ) ) ).

% combine_options_cases
tff(fact_384_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op167334234089630144_nat_o,Y3: option_nat] :
      ( ( ( Xb = none_num )
       => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y3) )
     => ( ( ( Y3 = none_nat )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y3) )
       => ( ! [A2: num,B3: nat] :
              ( ( Xb = some_num(A2) )
             => ( ( Y3 = some_nat(B3) )
               => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y3) ) )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y3) ) ) ) ).

% combine_options_cases
tff(fact_385_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op493924057549976310_num_o,Y3: option_num] :
      ( ( ( Xb = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y3) )
     => ( ( ( Y3 = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y3) )
       => ( ! [A2: num,B3: num] :
              ( ( Xb = some_num(A2) )
             => ( ( Y3 = some_num(B3) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y3) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y3) ) ) ) ).

% combine_options_cases
tff(fact_386_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_13)
    <=> ( 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_387_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_13: option_nat] : aa_option_nat_o(P,X_13)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X: nat] : aa_option_nat_o(P,some_nat(X)) ) ) ).

% split_option_all
tff(fact_388_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_13: option_num] : aa_option_num_o(P,X_13)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X: num] : aa_option_num_o(P,some_num(X)) ) ) ).

% split_option_all
tff(fact_389_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_13)
    <=> ( 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_390_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_13: option_nat] : aa_option_nat_o(P,X_13)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X: nat] : aa_option_nat_o(P,some_nat(X)) ) ) ).

% split_option_ex
tff(fact_391_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_13: option_num] : aa_option_num_o(P,X_13)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X: num] : aa_option_num_o(P,some_num(X)) ) ) ).

% split_option_ex
tff(fact_392_option_Oexhaust,axiom,
    ! [Y3: option4927543243414619207at_nat] :
      ( ( Y3 != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] : Y3 != some_P7363390416028606310at_nat(X22) ) ).

% option.exhaust
tff(fact_393_option_Oexhaust,axiom,
    ! [Y3: option_nat] :
      ( ( Y3 != none_nat )
     => ~ ! [X22: nat] : Y3 != some_nat(X22) ) ).

% option.exhaust
tff(fact_394_option_Oexhaust,axiom,
    ! [Y3: option_num] :
      ( ( Y3 != none_num )
     => ~ ! [X22: num] : Y3 != some_num(X22) ) ).

% option.exhaust
tff(fact_395_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X23: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X23) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_396_option_OdiscI,axiom,
    ! [Option: option_nat,X23: nat] :
      ( ( Option = some_nat(X23) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_397_option_OdiscI,axiom,
    ! [Option: option_num,X23: num] :
      ( ( Option = some_num(X23) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_398_option_Odistinct_I1_J,axiom,
    ! [X23: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X23) ).

% option.distinct(1)
tff(fact_399_option_Odistinct_I1_J,axiom,
    ! [X23: nat] : none_nat != some_nat(X23) ).

% option.distinct(1)
tff(fact_400_option_Odistinct_I1_J,axiom,
    ! [X23: num] : none_num != some_num(X23) ).

% option.distinct(1)
tff(fact_401_option_Osel,axiom,
    ! [X23: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X23)) = X23 ).

% option.sel
tff(fact_402_option_Osel,axiom,
    ! [X23: nat] : the_nat(some_nat(X23)) = X23 ).

% option.sel
tff(fact_403_option_Osel,axiom,
    ! [X23: num] : the_num(some_num(X23)) = X23 ).

% option.sel
tff(fact_404_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_405_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_406_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( the_num(Option) = the_num(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_407_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_408_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_409_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_410_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),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(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_411_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: 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))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_412_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_413_power__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y3: 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),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3) ) ) ).

% power_increasing_iff
tff(fact_414_power__increasing__iff,axiom,
    ! [B: rat,Xb: nat,Y3: 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),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3) ) ) ).

% power_increasing_iff
tff(fact_415_power__increasing__iff,axiom,
    ! [B: nat,Xb: nat,Y3: 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),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3) ) ) ).

% power_increasing_iff
tff(fact_416_power__increasing__iff,axiom,
    ! [B: int,Xb: nat,Y3: 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),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3) ) ) ).

% power_increasing_iff
tff(fact_417_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X2: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_418_insert__simp__norm,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = 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))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_419_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = 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))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_420_power__strict__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y3: 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),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3) ) ) ).

% power_strict_increasing_iff
tff(fact_421_power__strict__increasing__iff,axiom,
    ! [B: rat,Xb: nat,Y3: 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),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3) ) ) ).

% power_strict_increasing_iff
tff(fact_422_power__strict__increasing__iff,axiom,
    ! [B: nat,Xb: nat,Y3: 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),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3) ) ) ).

% power_strict_increasing_iff
tff(fact_423_power__strict__increasing__iff,axiom,
    ! [B: int,Xb: nat,Y3: 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),Y3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3) ) ) ).

% power_strict_increasing_iff
tff(fact_424_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)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),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,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_425_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)
       => ? [N: 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),N)),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,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_426_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(Ta,Xb)),Y3)
           => ( aa_nat_o(vEBT_vebt_member(Ta),Y3)
              | ( Xb = Y3 ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_427_power__inject__exp,axiom,
    ! [Aa: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_428_power__inject__exp,axiom,
    ! [Aa: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => ( ( aa_nat_rat(power_power_rat(Aa),Mb) = aa_nat_rat(power_power_rat(Aa),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_429_power__inject__exp,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_430_power__inject__exp,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Mb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_431_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_432_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ).

% min_Null_member
tff(fact_433_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList,S2) ) ).

% deg_SUcn_Node
tff(fact_434_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_435_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_436_dele__member__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_delete(Ta,Xb)),Y3)
      <=> ( ( Xb != Y3 )
          & aa_nat_o(vEBT_vebt_member(Ta),Y3) ) ) ) ).

% dele_member_cont_corr
tff(fact_437_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
      <=> aa_set_nat_o(member_nat(Xb),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_438_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% maxt_member
tff(fact_439_nat_Oinject,axiom,
    ! [X23: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X23) = aa_nat_nat(suc,Y22) )
    <=> ( X23 = Y22 ) ) ).

% nat.inject
tff(fact_440_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_441_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_442_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => ( aa_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_443_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y3)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
        & ! [Z2: nat] :
            ( ( aa_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,Y3),Z2) ) ) ) ).

% succ_member
tff(fact_444_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y3)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb)
        & ! [Z2: nat] :
            ( ( aa_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),Y3) ) ) ) ).

% pred_member
tff(fact_445_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = 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_446_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = 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_447_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_448_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Y3: 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,Y3),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) = Y3 ) ) ).

% high_inv
tff(fact_449_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Y3: 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,Y3),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_450_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_451_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_452_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_453_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_454_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_455_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_456_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(times_times_real(aa_num_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_457_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(times_times_rat(aa_num_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_458_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_459_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_460_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_461_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_462_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_463_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_464_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_465_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_466_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_467_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_468_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_eq
tff(fact_469_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc_right
tff(fact_470_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% Suc_le_mono
tff(fact_471_power__one__right,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),one_one_nat) = Aa ).

% power_one_right
tff(fact_472_power__one__right,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),one_one_nat) = Aa ).

% power_one_right
tff(fact_473_power__one__right,axiom,
    ! [Aa: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),one_one_nat) = Aa ).

% power_one_right
tff(fact_474_power__one__right,axiom,
    ! [Aa: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),one_one_nat) = Aa ).

% power_one_right
tff(fact_475_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = one_one_nat )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_476_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_477_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_Suc_Suc
tff(fact_478_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),B)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_479_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_480_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_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(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_481_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_482_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_483_distrib__right__numeral,axiom,
    ! [Aa: complex,B: complex,V: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_484_distrib__right__numeral,axiom,
    ! [Aa: real,B: real,V: num] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(Aa),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_485_distrib__right__numeral,axiom,
    ! [Aa: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_486_distrib__right__numeral,axiom,
    ! [Aa: 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,Aa),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,Aa),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_487_distrib__right__numeral,axiom,
    ! [Aa: 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,Aa),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,Aa),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_488_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = $ite(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)),numera1916890842035813515d_enat(V),numera1916890842035813515d_enat(U)) ).

% max_number_of(1)
tff(fact_489_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,U),aa_num_Code_integer(numera6620942414471956472nteger,V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,V),aa_num_Code_integer(numera6620942414471956472nteger,U)) ).

% max_number_of(1)
tff(fact_490_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(1)
tff(fact_491_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(1)
tff(fact_492_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_493_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_494_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(Xb)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(6)
tff(fact_495_max__0__1_I6_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Xb),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(6)
tff(fact_496_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,Xb)),one_one_real) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(6)
tff(fact_497_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(6)
tff(fact_498_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_499_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_500_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(5)
tff(fact_501_max__0__1_I5_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Xb)) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(5)
tff(fact_502_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(5)
tff(fact_503_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(5)
tff(fact_504_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_505_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_506_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc_right
tff(fact_507_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_508_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_509_le__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_510_le__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_511_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_512_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_513_less__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_514_less__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_515_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_516_power__add__numeral,axiom,
    ! [Aa: complex,Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_517_power__add__numeral,axiom,
    ! [Aa: real,Mb: num,Nb: num] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_518_power__add__numeral,axiom,
    ! [Aa: rat,Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_519_power__add__numeral,axiom,
    ! [Aa: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_520_power__add__numeral,axiom,
    ! [Aa: int,Mb: 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,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_521_power__add__numeral2,axiom,
    ! [Aa: complex,Mb: num,Nb: num,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_522_power__add__numeral2,axiom,
    ! [Aa: real,Mb: num,Nb: num,B: real] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_523_power__add__numeral2,axiom,
    ! [Aa: rat,Mb: num,Nb: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_524_power__add__numeral2,axiom,
    ! [Aa: nat,Mb: num,Nb: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_525_power__add__numeral2,axiom,
    ! [Aa: int,Mb: 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,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_526_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_527_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_528_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_529_div2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_530_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_531_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_532__C5_OIH_C_I1_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(treeList))
     => ( aa_nat_o(vEBT_invar_vebt(X3),na)
        & ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(X3),Xa)),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,X3))),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))))))))) ) ) ).

% "5.IH"(1)
tff(fact_533_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_534_power__Suc,axiom,
    ! [Aa: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(suc,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)) ).

% power_Suc
tff(fact_535_power__Suc,axiom,
    ! [Aa: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ).

% power_Suc
tff(fact_536_power__Suc,axiom,
    ! [Aa: rat,Nb: nat] : aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(Aa),Nb)) ).

% power_Suc
tff(fact_537_power__Suc,axiom,
    ! [Aa: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) ).

% power_Suc
tff(fact_538_power__Suc,axiom,
    ! [Aa: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ).

% power_Suc
tff(fact_539_power__Suc2,axiom,
    ! [Aa: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(suc,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)),Aa) ).

% power_Suc2
tff(fact_540_power__Suc2,axiom,
    ! [Aa: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),Aa) ).

% power_Suc2
tff(fact_541_power__Suc2,axiom,
    ! [Aa: rat,Nb: nat] : aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),Nb)),Aa) ).

% power_Suc2
tff(fact_542_power__Suc2,axiom,
    ! [Aa: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),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,Aa),Nb)),Aa) ).

% power_Suc2
tff(fact_543_power__Suc2,axiom,
    ! [Aa: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),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,Aa),Nb)),Aa) ).

% power_Suc2
tff(fact_544_Suc__inject,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Y3) )
     => ( Xb = Y3 ) ) ).

% Suc_inject
tff(fact_545_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_546_Suc__mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_547_nat__mult__max__left,axiom,
    ! [Mb: 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,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_max_left
tff(fact_548_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)) ).

% nat_mult_max_right
tff(fact_549_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),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,Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_550_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc
tff(fact_551_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),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,Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_552_power__commuting__commutes,axiom,
    ! [Xb: complex,Y3: complex,Nb: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Xb) )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_553_power__commuting__commutes,axiom,
    ! [Xb: real,Y3: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(Xb),Y3) = aa_real_real(times_times_real(Y3),Xb) )
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),Y3) = aa_real_real(times_times_real(Y3),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_554_power__commuting__commutes,axiom,
    ! [Xb: rat,Y3: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Y3) = aa_rat_rat(times_times_rat(Y3),Xb) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),Y3) = aa_rat_rat(times_times_rat(Y3),aa_nat_rat(power_power_rat(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_555_power__commuting__commutes,axiom,
    ! [Xb: nat,Y3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y3),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)),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_556_power__commuting__commutes,axiom,
    ! [Xb: int,Y3: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),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)),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_557_power__mult__distrib,axiom,
    ! [Aa: complex,B: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B)),Nb) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B),Nb)) ).

% power_mult_distrib
tff(fact_558_power__mult__distrib,axiom,
    ! [Aa: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(Aa),B)),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_mult_distrib
tff(fact_559_power__mult__distrib,axiom,
    ! [Aa: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(Aa),B)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_mult_distrib
tff(fact_560_power__mult__distrib,axiom,
    ! [Aa: 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,Aa),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).

% power_mult_distrib
tff(fact_561_power__mult__distrib,axiom,
    ! [Aa: 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,Aa),B)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ).

% power_mult_distrib
tff(fact_562_power__commutes,axiom,
    ! [Aa: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)),Aa) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)) ).

% power_commutes
tff(fact_563_power__commutes,axiom,
    ! [Aa: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),Aa) = aa_real_real(times_times_real(Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ).

% power_commutes
tff(fact_564_power__commutes,axiom,
    ! [Aa: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),Nb)),Aa) = aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(Aa),Nb)) ).

% power_commutes
tff(fact_565_power__commutes,axiom,
    ! [Aa: 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,Aa),Nb)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) ).

% power_commutes
tff(fact_566_power__commutes,axiom,
    ! [Aa: 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,Aa),Nb)),Aa) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ).

% power_commutes
tff(fact_567_power__mult,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Mb)),Nb) ).

% power_mult
tff(fact_568_power__mult,axiom,
    ! [Aa: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Mb)),Nb) ).

% power_mult
tff(fact_569_power__mult,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Mb)),Nb) ).

% power_mult
tff(fact_570_power__mult,axiom,
    ! [Aa: complex,Mb: nat,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Mb)),Nb) ).

% power_mult
tff(fact_571_power__odd__eq,axiom,
    ! [Aa: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_572_power__odd__eq,axiom,
    ! [Aa: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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(times_times_real(Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_573_power__odd__eq,axiom,
    ! [Aa: rat,Nb: nat] : aa_nat_rat(power_power_rat(Aa),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(Aa),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(Aa),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_574_power__odd__eq,axiom,
    ! [Aa: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_575_power__odd__eq,axiom,
    ! [Aa: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),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,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_576_nat__add__max__left,axiom,
    ! [Mb: 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,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q2)) ).

% nat_add_max_left
tff(fact_577_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ).

% nat_add_max_right
tff(fact_578_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_579_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_lessD
tff(fact_580_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_581_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( ( aa_nat_nat(suc,Mb) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_582_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_583_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_584_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
            & aa_nat_o(P,I5) ) ) ) ).

% Ex_less_Suc
tff(fact_585_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_586_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb)) ) ).

% not_less_eq
tff(fact_587_All__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
           => aa_nat_o(P,I5) ) ) ) ).

% All_less_Suc
tff(fact_588_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb)
    <=> ? [M3: nat] :
          ( ( Mb = aa_nat_nat(suc,M3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M3) ) ) ).

% Suc_less_eq2
tff(fact_589_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_590_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_591_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_592_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)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K3)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K3)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I2),K3) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_593_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)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I2))
               => aa_nat_o(P,I2) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_594_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_595_add__mult__distrib,axiom,
    ! [Mb: 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,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% add_mult_distrib
tff(fact_596_add__mult__distrib2,axiom,
    ! [K: nat,Mb: 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,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% add_mult_distrib2
tff(fact_597_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_598_nat__arith_Osuc1,axiom,
    ! [A: nat,K: nat,Aa: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Aa) )
     => ( aa_nat_nat(suc,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,Aa)) ) ) ).

% nat_arith.suc1
tff(fact_599_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc
tff(fact_600_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_601_le__cube,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb))) ).

% le_cube
tff(fact_602_le__square,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb)) ).

% le_square
tff(fact_603_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_604_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_605_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_606_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_leD
tff(fact_607_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_608_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_609_Suc__le__D,axiom,
    ! [Nb: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M4)
     => ? [M2: nat] : M4 = aa_nat_nat(suc,M2) ) ).

% Suc_le_D
tff(fact_610_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        | ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_611_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_612_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_613_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
             => aa_nat_o(P,M) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_614_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(P,Mb)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_615_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X2),X2)
       => ( ! [X2: nat,Y: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R,X2),Z3) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_616_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_617_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_618_div__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),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,Mb),Nb)),Q2) ).

% div_mult2_eq
tff(fact_619_left__right__inverse__power,axiom,
    ! [Xb: complex,Y3: complex,Nb: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y3) = one_one_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_620_left__right__inverse__power,axiom,
    ! [Xb: real,Y3: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(Xb),Y3) = one_one_real )
     => ( aa_real_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,Y3),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_621_left__right__inverse__power,axiom,
    ! [Xb: rat,Y3: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Y3) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Y3),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_622_left__right__inverse__power,axiom,
    ! [Xb: nat,Y3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y3) = 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,Y3),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_623_left__right__inverse__power,axiom,
    ! [Xb: int,Y3: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y3) = 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,Y3),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_624_power__add,axiom,
    ! [Aa: complex,Mb: nat,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)) ).

% power_add
tff(fact_625_power__add,axiom,
    ! [Aa: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ).

% power_add
tff(fact_626_power__add,axiom,
    ! [Aa: rat,Mb: nat,Nb: nat] : aa_nat_rat(power_power_rat(Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),Mb)),aa_nat_rat(power_power_rat(Aa),Nb)) ).

% power_add
tff(fact_627_power__add,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) ).

% power_add
tff(fact_628_power__add,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ).

% power_add
tff(fact_629_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_630_power__gt1__lemma,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb))) ) ).

% power_gt1_lemma
tff(fact_631_power__gt1__lemma,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(Aa),Nb))) ) ).

% power_gt1_lemma
tff(fact_632_power__gt1__lemma,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb))) ) ).

% power_gt1_lemma
tff(fact_633_power__gt1__lemma,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => 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,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb))) ) ).

% power_gt1_lemma
tff(fact_634_power__less__power__Suc,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),aa_real_real(times_times_real(Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb))) ) ).

% power_less_power_Suc
tff(fact_635_power__less__power__Suc,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),Nb)),aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(Aa),Nb))) ) ).

% power_less_power_Suc
tff(fact_636_power__less__power__Suc,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb))) ) ).

% power_less_power_Suc
tff(fact_637_power__less__power__Suc,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb))) ) ).

% power_less_power_Suc
tff(fact_638_power__gt1,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => 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,Aa),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_639_power__gt1,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_640_power__gt1,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => 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,Aa),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_641_power__gt1,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => 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,Aa),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_642_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_643_mult__numeral__1,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_644_mult__numeral__1,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_645_mult__numeral__1,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_646_mult__numeral__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_647_mult__numeral__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_648_mult__numeral__1__right,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),numera6690914467698888265omplex(one)) = Aa ).

% mult_numeral_1_right
tff(fact_649_mult__numeral__1__right,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,one)) = Aa ).

% mult_numeral_1_right
tff(fact_650_mult__numeral__1__right,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,one)) = Aa ).

% mult_numeral_1_right
tff(fact_651_mult__numeral__1__right,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_num_nat(numeral_numeral_nat,one)) = Aa ).

% mult_numeral_1_right
tff(fact_652_mult__numeral__1__right,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_num_int(numeral_numeral_int,one)) = Aa ).

% mult_numeral_1_right
tff(fact_653_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_654_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_655_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_656_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_657_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_658_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_659_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_660_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_661_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_662_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_663_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,Nb)),aa_nat_set_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_664_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_665_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_666_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_667_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_668_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N))),aa_nat_set_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_669_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_670_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_671_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_672_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_673_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).

% less_imp_Suc_add
tff(fact_674_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ? [K2: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)) ) ).

% less_iff_Suc_add
tff(fact_675_less__add__Suc2,axiom,
    ! [I: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),I))) ).

% less_add_Suc2
tff(fact_676_less__add__Suc1,axiom,
    ! [I: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb))) ).

% less_add_Suc1
tff(fact_677_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ~ ! [Q3: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)) ) ).

% less_natE
tff(fact_678_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_679_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_680_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_681_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_682_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_683_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)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_684_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)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_685_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_eq
tff(fact_686_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_687_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_688_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_689_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_690_less__mult__imp__div__less,axiom,
    ! [Mb: nat,I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),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,Mb),Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_691_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_692_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_693_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_694_power4__eq__xxxx,axiom,
    ! [Xb: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_695_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(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_696_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_697_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_698_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_699_power2__eq__square,axiom,
    ! [Aa: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),Aa) ).

% power2_eq_square
tff(fact_700_power2__eq__square,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(Aa),Aa) ).

% power2_eq_square
tff(fact_701_power2__eq__square,axiom,
    ! [Aa: rat] : aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(Aa),Aa) ).

% power2_eq_square
tff(fact_702_power2__eq__square,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Aa) ).

% power2_eq_square
tff(fact_703_power2__eq__square,axiom,
    ! [Aa: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Aa) ).

% power2_eq_square
tff(fact_704_power__even__eq,axiom,
    ! [Aa: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),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,Aa),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_705_power__even__eq,axiom,
    ! [Aa: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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,Aa),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_706_power__even__eq,axiom,
    ! [Aa: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),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,Aa),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_707_power__even__eq,axiom,
    ! [Aa: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_708_power3__eq__cube,axiom,
    ! [Aa: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),Aa)),Aa) ).

% power3_eq_cube
tff(fact_709_power3__eq__cube,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(times_times_real(aa_real_real(times_times_real(Aa),Aa)),Aa) ).

% power3_eq_cube
tff(fact_710_power3__eq__cube,axiom,
    ! [Aa: rat] : aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Aa),Aa)),Aa) ).

% power3_eq_cube
tff(fact_711_power3__eq__cube,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),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,Aa),Aa)),Aa) ).

% power3_eq_cube
tff(fact_712_power3__eq__cube,axiom,
    ! [Aa: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),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,Aa),Aa)),Aa) ).

% power3_eq_cube
tff(fact_713_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_714_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),Z) ).

% mult_2
tff(fact_715_mult__2,axiom,
    ! [Z: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(plus_plus_real(Z),Z) ).

% mult_2
tff(fact_716_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_717_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_718_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_719_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),Z) ).

% mult_2_right
tff(fact_720_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(times_times_real(Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(Z),Z) ).

% mult_2_right
tff(fact_721_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_722_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_723_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_724_left__add__twice,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Aa)),B) ).

% left_add_twice
tff(fact_725_left__add__twice,axiom,
    ! [Aa: real,B: real] : aa_real_real(plus_plus_real(Aa),aa_real_real(plus_plus_real(Aa),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Aa)),B) ).

% left_add_twice
tff(fact_726_left__add__twice,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(Aa),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Aa)),B) ).

% left_add_twice
tff(fact_727_left__add__twice,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),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)),B) ).

% left_add_twice
tff(fact_728_left__add__twice,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),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)),B) ).

% left_add_twice
tff(fact_729_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_730_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_731_power2__sum,axiom,
    ! [Xb: complex,Y3: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y3)) ).

% power2_sum
tff(fact_732_power2__sum,axiom,
    ! [Xb: real,Y3: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Xb),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y3)) ).

% power2_sum
tff(fact_733_power2__sum,axiom,
    ! [Xb: rat,Y3: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Xb),Y3)),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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y3)) ).

% power2_sum
tff(fact_734_power2__sum,axiom,
    ! [Xb: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y3)),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,Y3),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)),Y3)) ).

% power2_sum
tff(fact_735_power2__sum,axiom,
    ! [Xb: int,Y3: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y3)),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,Y3),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)),Y3)) ).

% power2_sum
tff(fact_736_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_737_power__divide,axiom,
    ! [Aa: complex,B: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)),Nb) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B),Nb)) ).

% power_divide
tff(fact_738_power__divide,axiom,
    ! [Aa: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(Aa),B)),Nb) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_divide
tff(fact_739_power__divide,axiom,
    ! [Aa: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(Aa),B)),Nb) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(Aa),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_divide
tff(fact_740_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X2: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_741_one__le__power,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ) ).

% one_le_power
tff(fact_742_one__le__power,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(Aa),Nb)) ) ).

% one_le_power
tff(fact_743_one__le__power,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Aa)
     => 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,Aa),Nb)) ) ).

% one_le_power
tff(fact_744_one__le__power,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Aa)
     => 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,Aa),Nb)) ) ).

% one_le_power
tff(fact_745_power__one__over,axiom,
    ! [Aa: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),Aa)),Nb) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)) ).

% power_one_over
tff(fact_746_power__one__over,axiom,
    ! [Aa: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),Aa)),Nb) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ).

% power_one_over
tff(fact_747_power__one__over,axiom,
    ! [Aa: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),Aa)),Nb) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(Aa),Nb)) ).

% power_one_over
tff(fact_748_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),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(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_749_power__less__imp__less__exp,axiom,
    ! [Aa: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_750_power__less__imp__less__exp,axiom,
    ! [Aa: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),Mb)),aa_nat_rat(power_power_rat(Aa),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_751_power__less__imp__less__exp,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_752_power__less__imp__less__exp,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_753_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),N3)) ) ) ).

% power_strict_increasing
tff(fact_754_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),Nb)),aa_nat_rat(power_power_rat(Aa),N3)) ) ) ).

% power_strict_increasing
tff(fact_755_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),N3)) ) ) ).

% power_strict_increasing
tff(fact_756_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),N3)) ) ) ).

% power_strict_increasing
tff(fact_757_power__increasing,axiom,
    ! [Nb: nat,N3: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),N3)) ) ) ).

% power_increasing
tff(fact_758_power__increasing,axiom,
    ! [Nb: nat,N3: nat,Aa: rat] :
      ( 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,one_one_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Nb)),aa_nat_rat(power_power_rat(Aa),N3)) ) ) ).

% power_increasing
tff(fact_759_power__increasing,axiom,
    ! [Nb: nat,N3: nat,Aa: nat] :
      ( 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,one_one_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),N3)) ) ) ).

% power_increasing
tff(fact_760_power__increasing,axiom,
    ! [Nb: nat,N3: nat,Aa: int] :
      ( 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,one_one_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),N3)) ) ) ).

% power_increasing
tff(fact_761_power__le__imp__le__exp,axiom,
    ! [Aa: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_762_power__le__imp__le__exp,axiom,
    ! [Aa: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Mb)),aa_nat_rat(power_power_rat(Aa),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_763_power__le__imp__le__exp,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_764_power__le__imp__le__exp,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_765_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_766_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_767_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_768_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_769_one__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_770_less__exp,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)) ).

% less_exp
tff(fact_771_self__le__ge2__pow,axiom,
    ! [K: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)) ) ).

% self_le_ge2_pow
tff(fact_772_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_773_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_774_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: 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))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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,Dega),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,Dega),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,Dega),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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_775__C5_OIH_C_I2_J,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),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,summary))),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))))))))) ).

% "5.IH"(2)
tff(fact_776_nested__mint,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Va: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Nb = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ 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,Va),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,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_777_sum__squares__bound,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y3)),aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_778_sum__squares__bound,axiom,
    ! [Xb: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y3)),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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_779_two__powr__height__bound__deg,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_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_780_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(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_781_div__exp__eq,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_782_div__exp__eq,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),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,Aa),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,Mb),Nb))) ).

% div_exp_eq
tff(fact_783__C10_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa)),
      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_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,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),
        $let(
          maxs: option_nat,
          maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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(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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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)))))) ))) ).

% "10"
tff(fact_784__C11_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa)),
      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,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
        $let(
          maxs: option_nat,
          maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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(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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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)))))) ))) ).

% "11"
tff(fact_785__C6_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa)),
      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_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,aa_num_num(bit1,one)))))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
        $let(
          sn: vEBT_VEBT,
          sn:= vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $ite(
              xa = 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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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)))))) )),
              one_one_nat )) ))) ).

% "6"
tff(fact_786__C8_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa)),
      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_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,aa_num_num(bit1,one)))))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          $ite(
            xa = 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,vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),
              $let(
                maxs: option_nat,
                maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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(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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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)))))) )),
            one_one_nat )))) ).

% "8"
tff(fact_787_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_788_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).

% minminNull
tff(fact_789_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ( vEBT_vebt_mint(Ta) = none_nat ) ) ).

% minNullmin
tff(fact_790_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% mint_member
tff(fact_791_height__compose__summary,axiom,
    ! [Summarya: vEBT_VEBT,Info: option4927543243414619207at_nat,Dega: 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,vEBT_Node(Info,Dega,TreeLista,Summarya))) ).

% height_compose_summary
tff(fact_792_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Mini) )
       => ( aa_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_793_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Xb) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% mint_corr
tff(fact_794_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_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) = some_nat(Xb) ) ) ) ).

% mint_sound
tff(fact_795_height__compose__child,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Dega: 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,vEBT_Node(Info,Dega,TreeLista,Summarya))) ) ).

% height_compose_child
tff(fact_796_bits__div__by__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),one_one_nat) = Aa ).

% bits_div_by_1
tff(fact_797_bits__div__by__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),one_one_int) = Aa ).

% bits_div_by_1
tff(fact_798_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( some_nat(Mb) = vEBT_vebt_mint(Ta) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% misiz
tff(fact_799_semiring__norm_I13_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% semiring_norm(13)
tff(fact_800_semiring__norm_I11_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Mb),one) = Mb ).

% semiring_norm(11)
tff(fact_801_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_802_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_803_power__mult__numeral,axiom,
    ! [Aa: nat,Mb: 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,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_804_power__mult__numeral,axiom,
    ! [Aa: real,Mb: 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,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_805_power__mult__numeral,axiom,
    ! [Aa: int,Mb: 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,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_806_power__mult__numeral,axiom,
    ! [Aa: complex,Mb: num,Nb: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_807_semiring__norm_I14_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_808_semiring__norm_I15_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),Nb)) ).

% semiring_norm(15)
tff(fact_809_semiring__norm_I16_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),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,Mb),Nb)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)))) ).

% semiring_norm(16)
tff(fact_810_del__x__not__mia,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: 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))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
               => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,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),
                            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,Dega),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 ))),Dega,newlist,sn) ),
                          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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Maa))),Dega,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_811_del__x__not__mi__new__node__nil,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: 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))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = 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,Dega),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 ))),Dega,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_812_del__x__not__mi,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: 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))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                   => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,H),
                            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,Dega),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 ))),Dega,Newlist,sn) ),
                          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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_813_del__x__mia,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: 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))),Dega)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,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,Dega),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,Dega),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,Dega),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),
                            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,Dega),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 ))),Dega,newlist,sn) ),
                          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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),Dega,newlist,Summarya) ) ) ),
                    vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_814_del__x__mi__lets__in__minNull,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: 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))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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,Dega),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,Dega),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,Dega),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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = 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,Dega),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 ))),Dega,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_815_del__x__mi__lets__in,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: 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))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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,Dega),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,Dega),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,Dega),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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,H),
                              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,Dega),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 ))),Dega,Newlist,sn) ),
                            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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_816_del__x__mi,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: 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))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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,Dega),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,Dega),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,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),
                              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,Dega),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 ))),Dega,newlist,sn) ),
                            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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Maa))),Dega,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_817_del__in__range,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: 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))),Dega)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,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,Dega),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,Dega),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,Dega),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),
                              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,Dega),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 ))),Dega,newlist,sn) ),
                            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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),Dega,newlist,Summarya) ) ) ),
                      vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_818__C9_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa)),
      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_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,aa_num_num(bit1,one)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),one_one_nat)),aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
        $ite(
          xa = 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,vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),
            $let(
              maxs: option_nat,
              maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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(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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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)))))) )),
          one_one_nat ))) ).

% "9"
tff(fact_819_real__arch__pow,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) ) ).

% real_arch_pow
tff(fact_820_less__eq__real__def,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% less_eq_real_def
tff(fact_821_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X3: real] : aa_set_real_o(member_real(X3),S3)
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_set_real_o(member_real(X2),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z4) )
       => ? [Y: real] :
            ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) )
            & ! [Z4: real] :
                ( ! [X2: real] :
                    ( aa_set_real_o(member_real(X2),S3)
                   => 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,Y),Z4) ) ) ) ) ).

% complete_real
tff(fact_822_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_823_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,m2),m2) ) ).

% numeral_code(2)
tff(fact_824_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(plus_plus_real(m2),m2) ) ).

% numeral_code(2)
tff(fact_825_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(m2),m2) ) ).

% numeral_code(2)
tff(fact_826_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_827_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_828_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_829_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_830_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_831_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_832_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_833_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,w),w) ) ).

% power_numeral_even
tff(fact_834_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(times_times_real(w),w) ) ).

% power_numeral_even
tff(fact_835_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_836_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_837_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_838_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_839_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(times_times_real(aa_real_real(times_times_real(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_840_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_841_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_842_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_843_four__x__squared,axiom,
    ! [Xb: real] : aa_real_real(times_times_real(aa_num_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(times_times_real(aa_num_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_844_L2__set__mult__ineq__lemma,axiom,
    ! [Aa: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(times_times_real(Aa),C))),aa_real_real(times_times_real(B),D))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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(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_845_div__mult2__numeral__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),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_846_div__mult2__numeral__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),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_847_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            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,Va)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_848__C3_C,axiom,
    aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa) = 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,aa_num_num(bit0,one)))))),
      $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList)),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
          $let(
            newnode: vEBT_VEBT,
            newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),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(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                    $let(
                      sn: vEBT_VEBT,
                      sn:= vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $ite(
                          xa = 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(xa = 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,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)) )) ) )),
        one_one_nat )) ).

% "3"
tff(fact_849__C2_C,axiom,
    aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa) = 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_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,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),one_one_nat)),one_one_nat)),
      $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,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,xa),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(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(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(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(
                              xa = 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(xa = 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 ) ) )) ).

% "2"
tff(fact_850__C5_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa)),
      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,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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,vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
            $let(
              sn: vEBT_VEBT,
              sn:= vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                $ite(
                  xa = 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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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)))))) )),
                  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(xa = 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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)) ))) ).

% "5"
tff(fact_851__C4_C,axiom,
    aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa) = 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_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,aa_num_num(bit0,one)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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,vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
          $let(
            sn: vEBT_VEBT,
            sn:= vEBT_vebt_delete(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
              $ite(
                xa = 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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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)))))) )),
                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(xa = 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(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),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_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)) )) ).

% "4"
tff(fact_852_vebt__delete_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_delete(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,Va)),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) ),
        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,Va)),TreeLista,Summarya),
        $ite(
          ( ( Xb = Mia )
          & ( Xb = Maa ) ),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),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,Va))),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,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(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,Va))),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),
                          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,Va))),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,Va)),newlist,sn) ),
                        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,Va))),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,Va)),newlist,Summarya) ) ) ),
                  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,Va)),TreeLista,Summarya) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_853_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_854__C1_C,axiom,
    aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa) = 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_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,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),
              $ite(xa = 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(xa = 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,deg),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)))))),xa),
        $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,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,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(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(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(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 ) ) ) )) ).

% "1"
tff(fact_855__C0_C,axiom,
    aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa) = 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,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
      $ite(
        ( ( xa = mi )
        & ( xa = 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(xa = 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(xa = 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,deg),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)))))),xa),
            $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,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,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(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(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(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 ) ) ) )) )) ).

% "0"
tff(fact_856_mintlistlength,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Maa)
          & ? [M2: 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_857_height__node,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,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,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya))) ) ).

% height_node
tff(fact_858_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_859_mul__shift,axiom,
    ! [Xb: nat,Y3: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y3) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xb)),some_nat(Y3)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_860_real__divide__square__eq,axiom,
    ! [R2: real,Aa: real] : aa_real_real(divide_divide_real(aa_real_real(times_times_real(R2),Aa)),aa_real_real(times_times_real(R2),R2)) = aa_real_real(divide_divide_real(Aa),R2) ).

% real_divide_square_eq
tff(fact_861_power__minus__is__div,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
     => ( aa_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,Aa),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))),Aa)),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_862_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_Suc_Suc
tff(fact_863_Suc__diff__diff,axiom,
    ! [Mb: 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,Mb)),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,Mb),Nb)),K) ).

% Suc_diff_diff
tff(fact_864_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_865_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_866_left__diff__distrib__numeral,axiom,
    ! [Aa: complex,B: complex,V: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_867_left__diff__distrib__numeral,axiom,
    ! [Aa: real,B: real,V: num] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(Aa),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(B),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_868_left__diff__distrib__numeral,axiom,
    ! [Aa: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Aa),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_869_left__diff__distrib__numeral,axiom,
    ! [Aa: 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,Aa),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,Aa),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_870_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),B)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_871_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_872_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_add__diff__add,axiom,
    ! [Aa: real,C: real,B: real,D: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),D)) = aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(Aa),B)),aa_real_real(minus_minus_real(C),D)) ).

% add_diff_add
tff(fact_882_add__diff__add,axiom,
    ! [Aa: rat,C: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(Aa),B)),aa_rat_rat(minus_minus_rat(C),D)) ).

% add_diff_add
tff(fact_883_add__diff__add,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).

% add_diff_add
tff(fact_884_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_885_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_886_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_887_Nat_Odiff__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% Nat.diff_cancel
tff(fact_888_diff__cancel2,axiom,
    ! [Mb: 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,Mb),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,Mb),Nb) ).

% diff_cancel2
tff(fact_889_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_890_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_891_eq__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( 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,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_892_le__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( 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,Mb),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,Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_893_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( 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,Mb),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,Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_894_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_895_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_896_le__diff__iff_H,axiom,
    ! [Aa: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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),Aa)),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),Aa) ) ) ) ).

% le_diff_iff'
tff(fact_897_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).

% diff_le_mono2
tff(fact_898_diff__mult__distrib,axiom,
    ! [Mb: 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,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% diff_mult_distrib
tff(fact_899_diff__mult__distrib2,axiom,
    ! [K: nat,Mb: 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,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% diff_mult_distrib2
tff(fact_900_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_901_mult__diff__mult,axiom,
    ! [Xb: real,Y3: real,Aa: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Y3)),aa_real_real(times_times_real(Aa),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),aa_real_real(minus_minus_real(Y3),B))),aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),Aa)),B)) ).

% mult_diff_mult
tff(fact_902_mult__diff__mult,axiom,
    ! [Xb: rat,Y3: rat,Aa: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Y3)),aa_rat_rat(times_times_rat(Aa),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),aa_rat_rat(minus_minus_rat(Y3),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Aa)),B)) ).

% mult_diff_mult
tff(fact_903_mult__diff__mult,axiom,
    ! [Xb: int,Y3: int,Aa: 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),Y3)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),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,Y3),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Aa)),B)) ).

% mult_diff_mult
tff(fact_904_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_905_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(suc,Mb)) ).

% diff_less_Suc
tff(fact_906_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_907_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_908_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_909_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_910_diff__less__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_911_less__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( 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,Mb),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,Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_912_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_913_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_914_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_915_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_916_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_917_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Mb) ).

% nat_minus_add_max
tff(fact_918_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,vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_919_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,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_920_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,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_921_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_922_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_923_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_924_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_925_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_926_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_927_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_928_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_929_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,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_930_power2__commute,axiom,
    ! [Xb: complex,Y3: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y3),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_931_power2__commute,axiom,
    ! [Xb: real,Y3: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xb),Y3)),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(minus_minus_real(Y3),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_932_power2__commute,axiom,
    ! [Xb: rat,Y3: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Y3),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_933_power2__commute,axiom,
    ! [Xb: int,Y3: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y3)),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,Y3),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_934_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_935_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_936_diff__le__diff__pow,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_937_power2__diff,axiom,
    ! [Xb: complex,Y3: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y3)) ).

% power2_diff
tff(fact_938_power2__diff,axiom,
    ! [Xb: real,Y3: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xb),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y3)) ).

% power2_diff
tff(fact_939_power2__diff,axiom,
    ! [Xb: rat,Y3: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y3)) ).

% power2_diff
tff(fact_940_power2__diff,axiom,
    ! [Xb: int,Y3: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y3)),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,Y3),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)),Y3)) ).

% power2_diff
tff(fact_941_vebt__delete_Osimps_I4_J,axiom,
    ! [Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya),Uu) = vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya) ).

% vebt_delete.simps(4)
tff(fact_942_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_943_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_944_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_945_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,vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_946_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_947_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,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_948_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,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(vEBT_T_d_e_l_e_t_e(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,Va)),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,Va))),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,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,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),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(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(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_949_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,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t(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,Va)),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,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(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(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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeLista,h)))),
                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),aa_nat_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_950_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,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r(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,Va)),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,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(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_951_le__add__diff__inverse,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
     => ( aa_real_real(plus_plus_real(B),aa_real_real(minus_minus_real(Aa),B)) = Aa ) ) ).

% le_add_diff_inverse
tff(fact_952_le__add__diff__inverse,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(Aa),B)) = Aa ) ) ).

% le_add_diff_inverse
tff(fact_953_le__add__diff__inverse,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),B)) = Aa ) ) ).

% le_add_diff_inverse
tff(fact_954_le__add__diff__inverse,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B)) = Aa ) ) ).

% le_add_diff_inverse
tff(fact_955_le__add__diff__inverse2,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
     => ( aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(Aa),B)),B) = Aa ) ) ).

% le_add_diff_inverse2
tff(fact_956_le__add__diff__inverse2,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(Aa),B)),B) = Aa ) ) ).

% le_add_diff_inverse2
tff(fact_957_le__add__diff__inverse2,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),B)),B) = Aa ) ) ).

% le_add_diff_inverse2
tff(fact_958_le__add__diff__inverse2,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B)),B) = Aa ) ) ).

% le_add_diff_inverse2
tff(fact_959_pred__less__length__list,axiom,
    ! [Dega: 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))),Dega)
     => ( 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,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,Dega),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,Dega),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,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),some_nat(Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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,Dega),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_960_pred__lesseq__max,axiom,
    ! [Dega: 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))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,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,Dega),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,Dega),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,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),some_nat(Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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,Dega),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_961_succ__greatereq__min,axiom,
    ! [Dega: 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))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,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,Dega),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,Dega),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,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= 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,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,Dega),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_962_succ__less__length__list,axiom,
    ! [Dega: 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))),Dega)
     => ( 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,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,Dega),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,Dega),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,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= 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,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,Dega),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_963_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,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t2(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,Va)),TreeLista,Summarya)),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),aa_nat_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_964_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_965_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Ux,Uy,Uz)) = some_nat(Maa) ).

% vebt_maxt.simps(3)
tff(fact_966_greater__shift,axiom,
    ! [Y3: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(Xb)),some_nat(Y3)) ) ).

% greater_shift
tff(fact_967_less__shift,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(Xb)),some_nat(Y3)) ) ).

% less_shift
tff(fact_968_add__shift,axiom,
    ! [Xb: nat,Y3: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y3) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xb)),some_nat(Y3)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_969_div__by__1,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),one_one_complex) = Aa ).

% div_by_1
tff(fact_970_div__by__1,axiom,
    ! [Aa: real] : aa_real_real(divide_divide_real(Aa),one_one_real) = Aa ).

% div_by_1
tff(fact_971_div__by__1,axiom,
    ! [Aa: rat] : aa_rat_rat(divide_divide_rat(Aa),one_one_rat) = Aa ).

% div_by_1
tff(fact_972_div__by__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),one_one_nat) = Aa ).

% div_by_1
tff(fact_973_div__by__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),one_one_int) = Aa ).

% div_by_1
tff(fact_974_linorder__neqE__linordered__idom,axiom,
    ! [Xb: real,Y3: real] :
      ( ( Xb != Y3 )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_975_linorder__neqE__linordered__idom,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ( Xb != Y3 )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_976_linorder__neqE__linordered__idom,axiom,
    ! [Xb: int,Y3: int] :
      ( ( Xb != Y3 )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_977_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(vEBT_T_i_n_s_e_r_t2(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_978_insersimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_12)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Ta),Y3)),one_one_nat) ) ) ).

% insersimp'
tff(fact_979_insertsimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( aa_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(vEBT_T_i_n_s_e_r_t2(Ta),L)),one_one_nat) ) ) ).

% insertsimp'
tff(fact_980_combine__common__factor,axiom,
    ! [Aa: real,E: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),E)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(plus_plus_real(Aa),B)),E)),C) ).

% combine_common_factor
tff(fact_981_combine__common__factor,axiom,
    ! [Aa: rat,E: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),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(Aa),B)),E)),C) ).

% combine_common_factor
tff(fact_982_combine__common__factor,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),E)),C) ).

% combine_common_factor
tff(fact_983_combine__common__factor,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),E)),C) ).

% combine_common_factor
tff(fact_984_distrib__right,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(Aa),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C)) ).

% distrib_right
tff(fact_985_distrib__right,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_986_distrib__right,axiom,
    ! [Aa: 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,Aa),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),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_987_distrib__right,axiom,
    ! [Aa: 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,Aa),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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_988_distrib__left,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(Aa),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),B)),aa_real_real(times_times_real(Aa),C)) ).

% distrib_left
tff(fact_989_distrib__left,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),B)),aa_rat_rat(times_times_rat(Aa),C)) ).

% distrib_left
tff(fact_990_distrib__left,axiom,
    ! [Aa: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),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),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)) ).

% distrib_left
tff(fact_991_distrib__left,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),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),B)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)) ).

% distrib_left
tff(fact_992_comm__semiring__class_Odistrib,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(Aa),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C)) ).

% comm_semiring_class.distrib
tff(fact_993_comm__semiring__class_Odistrib,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_994_comm__semiring__class_Odistrib,axiom,
    ! [Aa: 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,Aa),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),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_995_comm__semiring__class_Odistrib,axiom,
    ! [Aa: 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,Aa),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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_996_ring__class_Oring__distribs_I1_J,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(Aa),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),B)),aa_real_real(times_times_real(Aa),C)) ).

% ring_class.ring_distribs(1)
tff(fact_997_ring__class_Oring__distribs_I1_J,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),B)),aa_rat_rat(times_times_rat(Aa),C)) ).

% ring_class.ring_distribs(1)
tff(fact_998_ring__class_Oring__distribs_I1_J,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),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),B)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)) ).

% ring_class.ring_distribs(1)
tff(fact_999_ring__class_Oring__distribs_I2_J,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(Aa),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1000_ring__class_Oring__distribs_I2_J,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1001_ring__class_Oring__distribs_I2_J,axiom,
    ! [Aa: 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,Aa),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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1002_left__diff__distrib,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(Aa),B)),C) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C)) ).

% left_diff_distrib
tff(fact_1003_left__diff__distrib,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Aa),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_1004_left__diff__distrib,axiom,
    ! [Aa: 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,Aa),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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_1005_right__diff__distrib,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(Aa),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(Aa),B)),aa_real_real(times_times_real(Aa),C)) ).

% right_diff_distrib
tff(fact_1006_right__diff__distrib,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),B)),aa_rat_rat(times_times_rat(Aa),C)) ).

% right_diff_distrib
tff(fact_1007_right__diff__distrib,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),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),B)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)) ).

% right_diff_distrib
tff(fact_1008_left__diff__distrib_H,axiom,
    ! [B: real,C: real,Aa: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),C)),Aa) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(B),Aa)),aa_real_real(times_times_real(C),Aa)) ).

% left_diff_distrib'
tff(fact_1009_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,Aa: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C)),Aa) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),Aa)),aa_rat_rat(times_times_rat(C),Aa)) ).

% left_diff_distrib'
tff(fact_1010_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,Aa: 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)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)) ).

% left_diff_distrib'
tff(fact_1011_left__diff__distrib_H,axiom,
    ! [B: int,C: int,Aa: 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)),Aa) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)) ).

% left_diff_distrib'
tff(fact_1012_right__diff__distrib_H,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(Aa),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(Aa),B)),aa_real_real(times_times_real(Aa),C)) ).

% right_diff_distrib'
tff(fact_1013_right__diff__distrib_H,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),B)),aa_rat_rat(times_times_rat(Aa),C)) ).

% right_diff_distrib'
tff(fact_1014_right__diff__distrib_H,axiom,
    ! [Aa: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),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,Aa),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)) ).

% right_diff_distrib'
tff(fact_1015_right__diff__distrib_H,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),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),B)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)) ).

% right_diff_distrib'
tff(fact_1016_insert_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_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_1017_lambda__one,axiom,
    aTP_Lamm_af = aa_com3342990543769683525omplex(times_times_complex,one_one_complex) ).

% lambda_one
tff(fact_1018_lambda__one,axiom,
    aTP_Lamm_ag = times_times_real(one_one_real) ).

% lambda_one
tff(fact_1019_lambda__one,axiom,
    aTP_Lamm_ah = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_1020_lambda__one,axiom,
    aTP_Lamm_ai = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_1021_lambda__one,axiom,
    aTP_Lamm_aj = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_1022_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(vEBT_T_m_e_m_b_e_r(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_1023_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,Aa: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(Aa),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,Aa),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1024_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,Aa: num,B: num] : vEBT_V819420779217536731ft_num(F,some_num(Aa),some_num(B)) = some_num(aa_num_num(aa_num_fun_num_num(F,Aa),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1025_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,Aa: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(Aa)),some_nat(B)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,Aa),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1026_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_1027_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_num_fun_num_num,Uv: option_num] : vEBT_V819420779217536731ft_num(Uu,none_num,Uv) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1028_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_1029_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(vEBT_T_i_n_s_e_r_t(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_1030_insersimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_12)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(Ta),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insersimp
tff(fact_1031_insertsimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( aa_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(vEBT_T_i_n_s_e_r_t(Ta),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insertsimp
tff(fact_1032_less__1__mult,axiom,
    ! [Mb: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Mb)
     => ( 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(times_times_real(Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_1033_less__1__mult,axiom,
    ! [Mb: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Mb)
     => ( 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(Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_1034_less__1__mult,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Mb)
     => ( 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,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_1035_less__1__mult,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Mb)
     => ( 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,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_1036_add__mono1,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Aa),one_one_real)),aa_real_real(plus_plus_real(B),one_one_real)) ) ).

% add_mono1
tff(fact_1037_add__mono1,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).

% add_mono1
tff(fact_1038_add__mono1,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_1039_add__mono1,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_1040_less__add__one,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(plus_plus_real(Aa),one_one_real)) ).

% less_add_one
tff(fact_1041_less__add__one,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(plus_plus_rat(Aa),one_one_rat)) ).

% less_add_one
tff(fact_1042_less__add__one,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)) ).

% less_add_one
tff(fact_1043_less__add__one,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),one_one_int)) ).

% less_add_one
tff(fact_1044_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(plus_plus_real(I),K)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I),K)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1045_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,aa_rat_rat(minus_minus_rat(Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1046_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_1047_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_1048_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(plus_plus_real(I),K)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(minus_minus_real(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_1049_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),aa_rat_rat(minus_minus_rat(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_1050_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_1051_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_1052_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [Aa: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_real_real(plus_plus_real(B),aa_real_real(minus_minus_real(Aa),B)) = Aa ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1053_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [Aa: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(Aa),B)) = Aa ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1054_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [Aa: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),B)) = Aa ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1055_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [Aa: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B)) = Aa ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1056_eq__add__iff1,axiom,
    ! [Aa: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),E)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(Aa),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_1057_eq__add__iff1,axiom,
    ! [Aa: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),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(aa_rat_rat(minus_minus_rat(Aa),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_1058_eq__add__iff1,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_1059_eq__add__iff2,axiom,
    ! [Aa: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),E)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D) )
    <=> ( C = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),Aa)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_1060_eq__add__iff2,axiom,
    ! [Aa: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),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(aa_rat_rat(minus_minus_rat(B),Aa)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_1061_eq__add__iff2,axiom,
    ! [Aa: 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,Aa),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),Aa)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_1062_square__diff__square__factored,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y3),Y3)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Xb),Y3)),aa_real_real(minus_minus_real(Xb),Y3)) ).

% square_diff_square_factored
tff(fact_1063_square__diff__square__factored,axiom,
    ! [Xb: rat,Y3: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y3),Y3)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),Y3)),aa_rat_rat(minus_minus_rat(Xb),Y3)) ).

% square_diff_square_factored
tff(fact_1064_square__diff__square__factored,axiom,
    ! [Xb: int,Y3: 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,Y3),Y3)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y3)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y3)) ).

% square_diff_square_factored
tff(fact_1065_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y3 )
     => ( ( ( Xaa = none_P5556105721700978146at_nat )
         => ( Y3 != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xaa = some_P7363390416028606310at_nat(V2)
           => ( ( Xba = none_P5556105721700978146at_nat )
             => ( Y3 != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A2: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(A2) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xba = some_P7363390416028606310at_nat(B3) )
                   => ( Y3 != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A2),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1066_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y3: option_num] :
      ( ( vEBT_V819420779217536731ft_num(Xb,Xaa,Xba) = Y3 )
     => ( ( ( Xaa = none_num )
         => ( Y3 != none_num ) )
       => ( ( ? [V2: num] : Xaa = some_num(V2)
           => ( ( Xba = none_num )
             => ( Y3 != none_num ) ) )
         => ~ ! [A2: num] :
                ( ( Xaa = some_num(A2) )
               => ! [B3: num] :
                    ( ( Xba = some_num(B3) )
                   => ( Y3 != some_num(aa_num_num(aa_num_fun_num_num(Xb,A2),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1067_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y3: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y3 )
     => ( ( ( Xaa = none_nat )
         => ( Y3 != none_nat ) )
       => ( ( ? [V2: nat] : Xaa = some_nat(V2)
           => ( ( Xba = none_nat )
             => ( Y3 != none_nat ) ) )
         => ~ ! [A2: nat] :
                ( ( Xaa = some_nat(A2) )
               => ! [B3: nat] :
                    ( ( Xba = some_nat(B3) )
                   => ( Y3 != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xb,A2),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1068_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_1069_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw,some_num(V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1070_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),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1071_ordered__ring__class_Ole__add__iff2,axiom,
    ! [Aa: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),Aa)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1072_ordered__ring__class_Ole__add__iff2,axiom,
    ! [Aa: 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(Aa),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(aa_rat_rat(minus_minus_rat(B),Aa)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1073_ordered__ring__class_Ole__add__iff2,axiom,
    ! [Aa: 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,Aa),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),Aa)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1074_ordered__ring__class_Ole__add__iff1,axiom,
    ! [Aa: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(Aa),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1075_ordered__ring__class_Ole__add__iff1,axiom,
    ! [Aa: 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(Aa),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(aa_rat_rat(minus_minus_rat(Aa),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1076_ordered__ring__class_Ole__add__iff1,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1077_less__add__iff2,axiom,
    ! [Aa: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),Aa)),E)),D)) ) ).

% less_add_iff2
tff(fact_1078_less__add__iff2,axiom,
    ! [Aa: 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(Aa),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(aa_rat_rat(minus_minus_rat(B),Aa)),E)),D)) ) ).

% less_add_iff2
tff(fact_1079_less__add__iff2,axiom,
    ! [Aa: 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,Aa),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),Aa)),E)),D)) ) ).

% less_add_iff2
tff(fact_1080_less__add__iff1,axiom,
    ! [Aa: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(Aa),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_1081_less__add__iff1,axiom,
    ! [Aa: 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(Aa),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(aa_rat_rat(minus_minus_rat(Aa),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_1082_less__add__iff1,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_1083_square__diff__one__factored,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Xb)),one_one_complex) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),one_one_complex)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),one_one_complex)) ).

% square_diff_one_factored
tff(fact_1084_square__diff__one__factored,axiom,
    ! [Xb: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Xb)),one_one_real) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Xb),one_one_real)),aa_real_real(minus_minus_real(Xb),one_one_real)) ).

% square_diff_one_factored
tff(fact_1085_square__diff__one__factored,axiom,
    ! [Xb: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ).

% square_diff_one_factored
tff(fact_1086_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_1087_insert__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_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_1088_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1089_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1090_member__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_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_1091_vebt__mint_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Ux,Uy,Uz)) = some_nat(Mia) ).

% vebt_mint.simps(3)
tff(fact_1092_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_succ(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,Va)),TreeLista,Summarya),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
        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,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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),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,some_nat(l)),maxlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                  $let(
                    sc: option_nat,
                    sc:= 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,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)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_1093_vebt__pred_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_pred(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,Va)),TreeLista,Summarya),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
        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,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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),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,some_nat(l)),minlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                  $let(
                    pr: option_nat,
                    pr:= vEBT_vebt_pred(Summarya,h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),some_nat(Mia),none_nat),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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(TreeLista,the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_1094_real__average__minus__first,axiom,
    ! [Aa: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Aa),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Aa) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),Aa)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_1095_real__average__minus__second,axiom,
    ! [B: real,Aa: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(B),Aa)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Aa) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),Aa)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_1096_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,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(vEBT_T_s_u_c_c(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,Va)),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,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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              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,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(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeLista,h)),l)),
                          $let(
                            sc: option_nat,
                            sc:= 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(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_1097_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,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d(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,Va)),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,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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              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,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(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeLista,h)),l)),
                        $let(
                          pr: option_nat,
                          pr:= 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(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_1098_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,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(vEBT_T_s_u_c_c2(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,Va)),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,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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),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,some_nat(l)),maxlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_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(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_1099_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,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d2(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,Va)),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,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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),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,some_nat(l)),minlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_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(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_1100_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak(Ta),Xb)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_1101_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_al(Ta),Xb)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_1102_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_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_1103_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_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_1104_mult__commute__abs,axiom,
    ! [C: real] : aTP_Lamm_am(C) = times_times_real(C) ).

% mult_commute_abs
tff(fact_1105_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_an(C) = times_times_rat(C) ).

% mult_commute_abs
tff(fact_1106_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ao,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_1107_mult__commute__abs,axiom,
    ! [C: int] : aa_int_fun_int_int(aTP_Lamm_ap,C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_1108_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,Va: vEBT_VEBT,Vb: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_1109_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,Va: vEBT_VEBT,Vb: nat] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_1110_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,Va: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz)),Va) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_1111_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,Va: nat] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz)),Va) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_1112_pred__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_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_1113_succ_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_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_1114_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xb),Y3)
    <=> ( aa_set_nat_o(member_nat(Y3),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),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),Y3) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_1115_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xb),Y3)
    <=> ( aa_set_nat_o(member_nat(Y3),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
        & ! [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,Y3),X) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_1116_pred__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_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_1117_succ__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_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_1118_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz),Va) = none_nat ).

% vebt_succ.simps(3)
tff(fact_1119_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_1120_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1121_max__bot2,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(Xb,bot_bot_set_nat) = Xb ).

% max_bot2
tff(fact_1122_max__bot2,axiom,
    ! [Xb: set_int] : ord_max_set_int(Xb,bot_bot_set_int) = Xb ).

% max_bot2
tff(fact_1123_max__bot2,axiom,
    ! [Xb: set_real] : ord_max_set_real(Xb,bot_bot_set_real) = Xb ).

% max_bot2
tff(fact_1124_max__bot2,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),bot_bot_nat) = Xb ).

% max_bot2
tff(fact_1125_max__bot2,axiom,
    ! [Xb: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),bot_bo4199563552545308370d_enat) = Xb ).

% max_bot2
tff(fact_1126_max__bot,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(bot_bot_set_nat,Xb) = Xb ).

% max_bot
tff(fact_1127_max__bot,axiom,
    ! [Xb: set_int] : ord_max_set_int(bot_bot_set_int,Xb) = Xb ).

% max_bot
tff(fact_1128_max__bot,axiom,
    ! [Xb: set_real] : ord_max_set_real(bot_bot_set_real,Xb) = Xb ).

% max_bot
tff(fact_1129_max__bot,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xb) = Xb ).

% max_bot
tff(fact_1130_max__bot,axiom,
    ! [Xb: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,bot_bo4199563552545308370d_enat),Xb) = Xb ).

% max_bot
tff(fact_1131_max__less__iff__conj,axiom,
    ! [Xb: extended_enat,Y3: extended_enat,Z: extended_enat] :
      ( ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y3),Z)
    <=> ( ord_le72135733267957522d_enat(Xb,Z)
        & ord_le72135733267957522d_enat(Y3,Z) ) ) ).

% max_less_iff_conj
tff(fact_1132_max__less__iff__conj,axiom,
    ! [Xb: code_integer,Y3: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ord_max_Code_integer(Xb,Y3)),Z)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Z)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y3),Z) ) ) ).

% max_less_iff_conj
tff(fact_1133_max__less__iff__conj,axiom,
    ! [Xb: real,Y3: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(Xb),Y3)),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,Y3),Z) ) ) ).

% max_less_iff_conj
tff(fact_1134_max__less__iff__conj,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(Xb),Y3)),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,Y3),Z) ) ) ).

% max_less_iff_conj
tff(fact_1135_max__less__iff__conj,axiom,
    ! [Xb: num,Y3: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y3)),Z)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Z) ) ) ).

% max_less_iff_conj
tff(fact_1136_max__less__iff__conj,axiom,
    ! [Xb: nat,Y3: 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),Y3)),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,Y3),Z) ) ) ).

% max_less_iff_conj
tff(fact_1137_max__less__iff__conj,axiom,
    ! [Xb: int,Y3: 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),Y3)),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,Y3),Z) ) ) ).

% max_less_iff_conj
tff(fact_1138_max_Oabsorb4,axiom,
    ! [Aa: extended_enat,B: extended_enat] :
      ( ord_le72135733267957522d_enat(Aa,B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) = B ) ) ).

% max.absorb4
tff(fact_1139_max_Oabsorb4,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),B)
     => ( ord_max_Code_integer(Aa,B) = B ) ) ).

% max.absorb4
tff(fact_1140_max_Oabsorb4,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_real_real(ord_max_real(Aa),B) = B ) ) ).

% max.absorb4
tff(fact_1141_max_Oabsorb4,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_rat_rat(ord_max_rat(Aa),B) = B ) ) ).

% max.absorb4
tff(fact_1142_max_Oabsorb4,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) = B ) ) ).

% max.absorb4
tff(fact_1143_max_Oabsorb4,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) = B ) ) ).

% max.absorb4
tff(fact_1144_max_Oabsorb4,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) = B ) ) ).

% max.absorb4
tff(fact_1145_max_Oabsorb3,axiom,
    ! [B: extended_enat,Aa: extended_enat] :
      ( ord_le72135733267957522d_enat(B,Aa)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) = Aa ) ) ).

% max.absorb3
tff(fact_1146_max_Oabsorb3,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),Aa)
     => ( ord_max_Code_integer(Aa,B) = Aa ) ) ).

% max.absorb3
tff(fact_1147_max_Oabsorb3,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ( aa_real_real(ord_max_real(Aa),B) = Aa ) ) ).

% max.absorb3
tff(fact_1148_max_Oabsorb3,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ( aa_rat_rat(ord_max_rat(Aa),B) = Aa ) ) ).

% max.absorb3
tff(fact_1149_max_Oabsorb3,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) = Aa ) ) ).

% max.absorb3
tff(fact_1150_max_Oabsorb3,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) = Aa ) ) ).

% max.absorb3
tff(fact_1151_max_Oabsorb3,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) = Aa ) ) ).

% max.absorb3
tff(fact_1152_max_Oabsorb1,axiom,
    ! [B: extended_enat,Aa: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,Aa)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) = Aa ) ) ).

% max.absorb1
tff(fact_1153_max_Oabsorb1,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Aa)
     => ( ord_max_Code_integer(Aa,B) = Aa ) ) ).

% max.absorb1
tff(fact_1154_max_Oabsorb1,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( aa_rat_rat(ord_max_rat(Aa),B) = Aa ) ) ).

% max.absorb1
tff(fact_1155_max_Oabsorb1,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) = Aa ) ) ).

% max.absorb1
tff(fact_1156_max_Oabsorb1,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) = Aa ) ) ).

% max.absorb1
tff(fact_1157_max_Oabsorb1,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) = Aa ) ) ).

% max.absorb1
tff(fact_1158_max_Oabsorb2,axiom,
    ! [Aa: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(Aa,B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) = B ) ) ).

% max.absorb2
tff(fact_1159_max_Oabsorb2,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),B)
     => ( ord_max_Code_integer(Aa,B) = B ) ) ).

% max.absorb2
tff(fact_1160_max_Oabsorb2,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_rat_rat(ord_max_rat(Aa),B) = B ) ) ).

% max.absorb2
tff(fact_1161_max_Oabsorb2,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) = B ) ) ).

% max.absorb2
tff(fact_1162_max_Oabsorb2,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) = B ) ) ).

% max.absorb2
tff(fact_1163_max_Oabsorb2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) = B ) ) ).

% max.absorb2
tff(fact_1164_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,Aa: extended_enat] :
      ( ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C),Aa)
    <=> ( ord_le2932123472753598470d_enat(B,Aa)
        & ord_le2932123472753598470d_enat(C,Aa) ) ) ).

% max.bounded_iff
tff(fact_1165_max_Obounded__iff,axiom,
    ! [B: code_integer,C: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),Aa)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Aa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_1166_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),Aa)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_1167_max_Obounded__iff,axiom,
    ! [B: num,C: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),Aa)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_1168_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,Aa: 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)),Aa)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_1169_max_Obounded__iff,axiom,
    ! [B: int,C: int,Aa: 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)),Aa)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_1170_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_1171_dual__order_Orefl,axiom,
    ! [Aa: set_int] : aa_set_int_o(ord_less_eq_set_int(Aa),Aa) ).

% dual_order.refl
tff(fact_1172_dual__order_Orefl,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Aa) ).

% dual_order.refl
tff(fact_1173_dual__order_Orefl,axiom,
    ! [Aa: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Aa) ).

% dual_order.refl
tff(fact_1174_dual__order_Orefl,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Aa) ).

% dual_order.refl
tff(fact_1175_dual__order_Orefl,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Aa) ).

% dual_order.refl
tff(fact_1176_order__refl,axiom,
    ! [Xb: set_int] : aa_set_int_o(ord_less_eq_set_int(Xb),Xb) ).

% order_refl
tff(fact_1177_order__refl,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Xb) ).

% order_refl
tff(fact_1178_order__refl,axiom,
    ! [Xb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Xb) ).

% order_refl
tff(fact_1179_order__refl,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Xb) ).

% order_refl
tff(fact_1180_order__refl,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xb) ).

% order_refl
tff(fact_1181_max_Oidem,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Aa) = Aa ).

% max.idem
tff(fact_1182_max_Oidem,axiom,
    ! [Aa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),Aa) = Aa ).

% max.idem
tff(fact_1183_max_Oidem,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),Aa) = Aa ).

% max.idem
tff(fact_1184_max_Oidem,axiom,
    ! [Aa: code_integer] : ord_max_Code_integer(Aa,Aa) = Aa ).

% max.idem
tff(fact_1185_max_Oleft__idem,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) ).

% max.left_idem
tff(fact_1186_max_Oleft__idem,axiom,
    ! [Aa: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) ).

% max.left_idem
tff(fact_1187_max_Oleft__idem,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) ).

% max.left_idem
tff(fact_1188_max_Oleft__idem,axiom,
    ! [Aa: code_integer,B: code_integer] : ord_max_Code_integer(Aa,ord_max_Code_integer(Aa,B)) = ord_max_Code_integer(Aa,B) ).

% max.left_idem
tff(fact_1189_max_Oright__idem,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) ).

% max.right_idem
tff(fact_1190_max_Oright__idem,axiom,
    ! [Aa: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) ).

% max.right_idem
tff(fact_1191_max_Oright__idem,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B)),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) ).

% max.right_idem
tff(fact_1192_max_Oright__idem,axiom,
    ! [Aa: code_integer,B: code_integer] : ord_max_Code_integer(ord_max_Code_integer(Aa,B),B) = ord_max_Code_integer(Aa,B) ).

% max.right_idem
tff(fact_1193_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_1194_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y3: extended_enat,Xb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,Y3)
     => ( plus_p3455044024723400733d_enat(Xb,minus_3235023915231533773d_enat(Y3,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Y3),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1195_order__antisym__conv,axiom,
    ! [Y3: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y3),Xb)
     => ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_1196_order__antisym__conv,axiom,
    ! [Y3: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_1197_order__antisym__conv,axiom,
    ! [Y3: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_1198_order__antisym__conv,axiom,
    ! [Y3: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_1199_order__antisym__conv,axiom,
    ! [Y3: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_1200_linorder__le__cases,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb) ) ).

% linorder_le_cases
tff(fact_1201_linorder__le__cases,axiom,
    ! [Xb: num,Y3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb) ) ).

% linorder_le_cases
tff(fact_1202_linorder__le__cases,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb) ) ).

% linorder_le_cases
tff(fact_1203_linorder__le__cases,axiom,
    ! [Xb: int,Y3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb) ) ).

% linorder_le_cases
tff(fact_1204_ord__le__eq__subst,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1205_ord__le__eq__subst,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1206_ord__le__eq__subst,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1207_ord__le__eq__subst,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1208_ord__le__eq__subst,axiom,
    ! [Aa: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1209_ord__le__eq__subst,axiom,
    ! [Aa: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1210_ord__le__eq__subst,axiom,
    ! [Aa: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1211_ord__le__eq__subst,axiom,
    ! [Aa: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1212_ord__le__eq__subst,axiom,
    ! [Aa: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1213_ord__le__eq__subst,axiom,
    ! [Aa: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1214_ord__eq__le__subst,axiom,
    ! [Aa: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( Aa = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1215_ord__eq__le__subst,axiom,
    ! [Aa: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( Aa = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1216_ord__eq__le__subst,axiom,
    ! [Aa: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( Aa = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1217_ord__eq__le__subst,axiom,
    ! [Aa: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( Aa = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1218_ord__eq__le__subst,axiom,
    ! [Aa: rat,F: fun_num_rat,B: num,C: num] :
      ( ( Aa = aa_num_rat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1219_ord__eq__le__subst,axiom,
    ! [Aa: num,F: fun_num_num,B: num,C: num] :
      ( ( Aa = aa_num_num(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_num_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1220_ord__eq__le__subst,axiom,
    ! [Aa: nat,F: fun_num_nat,B: num,C: num] :
      ( ( Aa = aa_num_nat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_num_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1221_ord__eq__le__subst,axiom,
    ! [Aa: int,F: fun_num_int,B: num,C: num] :
      ( ( Aa = aa_num_int(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_num_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1222_ord__eq__le__subst,axiom,
    ! [Aa: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( Aa = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1223_ord__eq__le__subst,axiom,
    ! [Aa: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( Aa = aa_nat_num(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_nat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1224_linorder__linear,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb) ) ).

% linorder_linear
tff(fact_1225_linorder__linear,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb) ) ).

% linorder_linear
tff(fact_1226_linorder__linear,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb) ) ).

% linorder_linear
tff(fact_1227_linorder__linear,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb) ) ).

% linorder_linear
tff(fact_1228_order__eq__refl,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( ( Xb = Y3 )
     => aa_set_int_o(ord_less_eq_set_int(Xb),Y3) ) ).

% order_eq_refl
tff(fact_1229_order__eq__refl,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ( Xb = Y3 )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3) ) ).

% order_eq_refl
tff(fact_1230_order__eq__refl,axiom,
    ! [Xb: num,Y3: num] :
      ( ( Xb = Y3 )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3) ) ).

% order_eq_refl
tff(fact_1231_order__eq__refl,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ( Xb = Y3 )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3) ) ).

% order_eq_refl
tff(fact_1232_order__eq__refl,axiom,
    ! [Xb: int,Y3: int] :
      ( ( Xb = Y3 )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3) ) ).

% order_eq_refl
tff(fact_1233_order__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1234_order__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1235_order__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1236_order__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1237_order__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1238_order__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1239_order__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1240_order__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1241_order__subst2,axiom,
    ! [Aa: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1242_order__subst2,axiom,
    ! [Aa: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_1243_order__subst1,axiom,
    ! [Aa: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1244_order__subst1,axiom,
    ! [Aa: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1245_order__subst1,axiom,
    ! [Aa: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1246_order__subst1,axiom,
    ! [Aa: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1247_order__subst1,axiom,
    ! [Aa: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_rat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_1248_order__subst1,axiom,
    ! [Aa: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_num_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_1249_order__subst1,axiom,
    ! [Aa: num,F: fun_nat_num,B: nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_nat_num(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_nat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_1250_order__subst1,axiom,
    ! [Aa: num,F: fun_int_num,B: int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_int_num(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X2)),aa_int_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_int_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_1251_order__subst1,axiom,
    ! [Aa: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1252_order__subst1,axiom,
    ! [Aa: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_num_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1253_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: set_int,B: set_int] :
      ( ( Aa = B )
    <=> ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
        & aa_set_int_o(ord_less_eq_set_int(B),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_1254_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_1255_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: num,B: num] :
      ( ( Aa = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_1256_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_1257_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_1258_antisym,axiom,
    ! [Aa: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),Aa)
       => ( Aa = B ) ) ) ).

% antisym
tff(fact_1259_antisym,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
       => ( Aa = B ) ) ) ).

% antisym
tff(fact_1260_antisym,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
       => ( Aa = B ) ) ) ).

% antisym
tff(fact_1261_antisym,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
       => ( Aa = B ) ) ) ).

% antisym
tff(fact_1262_antisym,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
       => ( Aa = B ) ) ) ).

% antisym
tff(fact_1263_dual__order_Otrans,axiom,
    ! [B: set_int,Aa: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),Aa)
     => ( aa_set_int_o(ord_less_eq_set_int(C),B)
       => aa_set_int_o(ord_less_eq_set_int(C),Aa) ) ) ).

% dual_order.trans
tff(fact_1264_dual__order_Otrans,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa) ) ) ).

% dual_order.trans
tff(fact_1265_dual__order_Otrans,axiom,
    ! [B: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa) ) ) ).

% dual_order.trans
tff(fact_1266_dual__order_Otrans,axiom,
    ! [B: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
     => ( 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) ) ) ).

% dual_order.trans
tff(fact_1267_dual__order_Otrans,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( 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) ) ) ).

% dual_order.trans
tff(fact_1268_dual__order_Oantisym,axiom,
    ! [B: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),Aa)
     => ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
       => ( Aa = B ) ) ) ).

% dual_order.antisym
tff(fact_1269_dual__order_Oantisym,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
       => ( Aa = B ) ) ) ).

% dual_order.antisym
tff(fact_1270_dual__order_Oantisym,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
       => ( Aa = B ) ) ) ).

% dual_order.antisym
tff(fact_1271_dual__order_Oantisym,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
       => ( Aa = B ) ) ) ).

% dual_order.antisym
tff(fact_1272_dual__order_Oantisym,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
       => ( Aa = B ) ) ) ).

% dual_order.antisym
tff(fact_1273_dual__order_Oeq__iff,axiom,
    ! [Aa: set_int,B: set_int] :
      ( ( Aa = B )
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),Aa)
        & aa_set_int_o(ord_less_eq_set_int(Aa),B) ) ) ).

% dual_order.eq_iff
tff(fact_1274_dual__order_Oeq__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ) ).

% dual_order.eq_iff
tff(fact_1275_dual__order_Oeq__iff,axiom,
    ! [Aa: num,B: num] :
      ( ( Aa = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B) ) ) ).

% dual_order.eq_iff
tff(fact_1276_dual__order_Oeq__iff,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B) ) ) ).

% dual_order.eq_iff
tff(fact_1277_dual__order_Oeq__iff,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B) ) ) ).

% dual_order.eq_iff
tff(fact_1278_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Aa: rat,B: rat] :
      ( ! [A2: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P,A2),B3) )
     => ( ! [A2: rat,B3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(P,A2),B3) )
       => aa_rat_o(aa_rat_fun_rat_o(P,Aa),B) ) ) ).

% linorder_wlog
tff(fact_1279_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,Aa: num,B: num] :
      ( ! [A2: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B3)
         => aa_num_o(aa_num_fun_num_o(P,A2),B3) )
     => ( ! [A2: num,B3: num] :
            ( aa_num_o(aa_num_fun_num_o(P,B3),A2)
           => aa_num_o(aa_num_fun_num_o(P,A2),B3) )
       => aa_num_o(aa_num_fun_num_o(P,Aa),B) ) ) ).

% linorder_wlog
tff(fact_1280_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,Aa: nat,B: nat] :
      ( ! [A2: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B3) )
     => ( ! [A2: nat,B3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(P,A2),B3) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Aa),B) ) ) ).

% linorder_wlog
tff(fact_1281_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,Aa: int,B: int] :
      ( ! [A2: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B3)
         => aa_int_o(aa_int_fun_int_o(P,A2),B3) )
     => ( ! [A2: int,B3: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B3),A2)
           => aa_int_o(aa_int_fun_int_o(P,A2),B3) )
       => aa_int_o(aa_int_fun_int_o(P,Aa),B) ) ) ).

% linorder_wlog
tff(fact_1282_order__trans,axiom,
    ! [Xb: set_int,Y3: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
     => ( aa_set_int_o(ord_less_eq_set_int(Y3),Z)
       => aa_set_int_o(ord_less_eq_set_int(Xb),Z) ) ) ).

% order_trans
tff(fact_1283_order__trans,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) ) ) ).

% order_trans
tff(fact_1284_order__trans,axiom,
    ! [Xb: num,Y3: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) ) ) ).

% order_trans
tff(fact_1285_order__trans,axiom,
    ! [Xb: nat,Y3: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) ) ) ).

% order_trans
tff(fact_1286_order__trans,axiom,
    ! [Xb: int,Y3: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) ) ) ).

% order_trans
tff(fact_1287_order_Otrans,axiom,
    ! [Aa: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => aa_set_int_o(ord_less_eq_set_int(Aa),C) ) ) ).

% order.trans
tff(fact_1288_order_Otrans,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),C) ) ) ).

% order.trans
tff(fact_1289_order_Otrans,axiom,
    ! [Aa: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),C) ) ) ).

% order.trans
tff(fact_1290_order_Otrans,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),C) ) ) ).

% order.trans
tff(fact_1291_order_Otrans,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),C) ) ) ).

% order.trans
tff(fact_1292_order__antisym,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
     => ( aa_set_int_o(ord_less_eq_set_int(Y3),Xb)
       => ( Xb = Y3 ) ) ) ).

% order_antisym
tff(fact_1293_order__antisym,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb)
       => ( Xb = Y3 ) ) ) ).

% order_antisym
tff(fact_1294_order__antisym,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb)
       => ( Xb = Y3 ) ) ) ).

% order_antisym
tff(fact_1295_order__antisym,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb)
       => ( Xb = Y3 ) ) ) ).

% order_antisym
tff(fact_1296_order__antisym,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb)
       => ( Xb = Y3 ) ) ) ).

% order_antisym
tff(fact_1297_ord__le__eq__trans,axiom,
    ! [Aa: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
     => ( ( B = C )
       => aa_set_int_o(ord_less_eq_set_int(Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_1298_ord__le__eq__trans,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_1299_ord__le__eq__trans,axiom,
    ! [Aa: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_1300_ord__le__eq__trans,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_1301_ord__le__eq__trans,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_1302_ord__eq__le__trans,axiom,
    ! [Aa: set_int,B: set_int,C: set_int] :
      ( ( Aa = B )
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => aa_set_int_o(ord_less_eq_set_int(Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_1303_ord__eq__le__trans,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( ( Aa = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_1304_ord__eq__le__trans,axiom,
    ! [Aa: num,B: num,C: num] :
      ( ( Aa = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_1305_ord__eq__le__trans,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( ( Aa = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_1306_ord__eq__le__trans,axiom,
    ! [Aa: int,B: int,C: int] :
      ( ( Aa = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_1307_order__class_Oorder__eq__iff,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( ( Xb = Y3 )
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
        & aa_set_int_o(ord_less_eq_set_int(Y3),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1308_order__class_Oorder__eq__iff,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ( Xb = Y3 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1309_order__class_Oorder__eq__iff,axiom,
    ! [Xb: num,Y3: num] :
      ( ( Xb = Y3 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1310_order__class_Oorder__eq__iff,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ( Xb = Y3 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1311_order__class_Oorder__eq__iff,axiom,
    ! [Xb: int,Y3: int] :
      ( ( Xb = Y3 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1312_le__cases3,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y3) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y3)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1313_le__cases3,axiom,
    ! [Xb: num,Y3: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y3) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y3)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Z)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1314_le__cases3,axiom,
    ! [Xb: nat,Y3: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y3) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y3)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1315_le__cases3,axiom,
    ! [Xb: int,Y3: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y3) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y3)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1316_nle__le,axiom,
    ! [Aa: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
        & ( B != Aa ) ) ) ).

% nle_le
tff(fact_1317_nle__le,axiom,
    ! [Aa: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
        & ( B != Aa ) ) ) ).

% nle_le
tff(fact_1318_nle__le,axiom,
    ! [Aa: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
        & ( B != Aa ) ) ) ).

% nle_le
tff(fact_1319_nle__le,axiom,
    ! [Aa: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
        & ( B != Aa ) ) ) ).

% nle_le
tff(fact_1320_lt__ex,axiom,
    ! [Xb: real] :
    ? [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ).

% lt_ex
tff(fact_1321_lt__ex,axiom,
    ! [Xb: rat] :
    ? [Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ).

% lt_ex
tff(fact_1322_lt__ex,axiom,
    ! [Xb: int] :
    ? [Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ).

% lt_ex
tff(fact_1323_gt__ex,axiom,
    ! [Xb: real] :
    ? [X_12: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X_12) ).

% gt_ex
tff(fact_1324_gt__ex,axiom,
    ! [Xb: rat] :
    ? [X_12: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),X_12) ).

% gt_ex
tff(fact_1325_gt__ex,axiom,
    ! [Xb: nat] :
    ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X_12) ).

% gt_ex
tff(fact_1326_gt__ex,axiom,
    ! [Xb: int] :
    ? [X_12: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),X_12) ).

% gt_ex
tff(fact_1327_dense,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Y3) ) ) ).

% dense
tff(fact_1328_dense,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ? [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z3)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),Y3) ) ) ).

% dense
tff(fact_1329_less__imp__neq,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% less_imp_neq
tff(fact_1330_less__imp__neq,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% less_imp_neq
tff(fact_1331_less__imp__neq,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% less_imp_neq
tff(fact_1332_less__imp__neq,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% less_imp_neq
tff(fact_1333_less__imp__neq,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% less_imp_neq
tff(fact_1334_order_Oasym,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa) ) ).

% order.asym
tff(fact_1335_order_Oasym,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa) ) ).

% order.asym
tff(fact_1336_order_Oasym,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa) ) ).

% order.asym
tff(fact_1337_order_Oasym,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa) ) ).

% order.asym
tff(fact_1338_order_Oasym,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa) ) ).

% order.asym
tff(fact_1339_ord__eq__less__trans,axiom,
    ! [Aa: real,B: real,C: real] :
      ( ( Aa = B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_1340_ord__eq__less__trans,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( ( Aa = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_1341_ord__eq__less__trans,axiom,
    ! [Aa: num,B: num,C: num] :
      ( ( Aa = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_1342_ord__eq__less__trans,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( ( Aa = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_1343_ord__eq__less__trans,axiom,
    ! [Aa: int,B: int,C: int] :
      ( ( Aa = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_1344_ord__less__eq__trans,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ( B = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_1345_ord__less__eq__trans,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_1346_ord__less__eq__trans,axiom,
    ! [Aa: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_1347_ord__less__eq__trans,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_1348_ord__less__eq__trans,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_1349_less__induct,axiom,
    ! [P: fun_nat_o,Aa: nat] :
      ( ! [X2: nat] :
          ( ! [Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X2)
             => aa_nat_o(P,Y2) )
         => aa_nat_o(P,X2) )
     => aa_nat_o(P,Aa) ) ).

% less_induct
tff(fact_1350_antisym__conv3,axiom,
    ! [Y3: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv3
tff(fact_1351_antisym__conv3,axiom,
    ! [Y3: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv3
tff(fact_1352_antisym__conv3,axiom,
    ! [Y3: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv3
tff(fact_1353_antisym__conv3,axiom,
    ! [Y3: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv3
tff(fact_1354_antisym__conv3,axiom,
    ! [Y3: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv3
tff(fact_1355_linorder__cases,axiom,
    ! [Xb: real,Y3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ( ( Xb != Y3 )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ) ).

% linorder_cases
tff(fact_1356_linorder__cases,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ( ( Xb != Y3 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ) ).

% linorder_cases
tff(fact_1357_linorder__cases,axiom,
    ! [Xb: num,Y3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ( ( Xb != Y3 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb) ) ) ).

% linorder_cases
tff(fact_1358_linorder__cases,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ( ( Xb != Y3 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ) ).

% linorder_cases
tff(fact_1359_linorder__cases,axiom,
    ! [Xb: int,Y3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ( ( Xb != Y3 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ) ).

% linorder_cases
tff(fact_1360_dual__order_Oasym,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ).

% dual_order.asym
tff(fact_1361_dual__order_Oasym,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ).

% dual_order.asym
tff(fact_1362_dual__order_Oasym,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B) ) ).

% dual_order.asym
tff(fact_1363_dual__order_Oasym,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B) ) ).

% dual_order.asym
tff(fact_1364_dual__order_Oasym,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B) ) ).

% dual_order.asym
tff(fact_1365_dual__order_Oirrefl,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Aa) ).

% dual_order.irrefl
tff(fact_1366_dual__order_Oirrefl,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Aa) ).

% dual_order.irrefl
tff(fact_1367_dual__order_Oirrefl,axiom,
    ! [Aa: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Aa) ).

% dual_order.irrefl
tff(fact_1368_dual__order_Oirrefl,axiom,
    ! [Aa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Aa) ).

% dual_order.irrefl
tff(fact_1369_dual__order_Oirrefl,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Aa) ).

% dual_order.irrefl
tff(fact_1370_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
    <=> ? [N4: nat] :
          ( aa_nat_o(P,N4)
          & ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N4)
             => ~ aa_nat_o(P,M5) ) ) ) ).

% exists_least_iff
tff(fact_1371_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,Aa: real,B: real] :
      ( ! [A2: real,B3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B3)
         => aa_real_o(aa_real_fun_real_o(P,A2),B3) )
     => ( ! [A2: real] : aa_real_o(aa_real_fun_real_o(P,A2),A2)
       => ( ! [A2: real,B3: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B3),A2)
             => aa_real_o(aa_real_fun_real_o(P,A2),B3) )
         => aa_real_o(aa_real_fun_real_o(P,Aa),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1372_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Aa: rat,B: rat] :
      ( ! [A2: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P,A2),B3) )
     => ( ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A2),A2)
       => ( ! [A2: rat,B3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(P,A2),B3) )
         => aa_rat_o(aa_rat_fun_rat_o(P,Aa),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1373_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,Aa: num,B: num] :
      ( ! [A2: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B3)
         => aa_num_o(aa_num_fun_num_o(P,A2),B3) )
     => ( ! [A2: num] : aa_num_o(aa_num_fun_num_o(P,A2),A2)
       => ( ! [A2: num,B3: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B3),A2)
             => aa_num_o(aa_num_fun_num_o(P,A2),B3) )
         => aa_num_o(aa_num_fun_num_o(P,Aa),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1374_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,Aa: nat,B: nat] :
      ( ! [A2: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B3) )
     => ( ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A2),A2)
       => ( ! [A2: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(P,A2),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Aa),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1375_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,Aa: int,B: int] :
      ( ! [A2: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B3)
         => aa_int_o(aa_int_fun_int_o(P,A2),B3) )
     => ( ! [A2: int] : aa_int_o(aa_int_fun_int_o(P,A2),A2)
       => ( ! [A2: int,B3: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B3),A2)
             => aa_int_o(aa_int_fun_int_o(P,A2),B3) )
         => aa_int_o(aa_int_fun_int_o(P,Aa),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1376_order_Ostrict__trans,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C) ) ) ).

% order.strict_trans
tff(fact_1377_order_Ostrict__trans,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% order.strict_trans
tff(fact_1378_order_Ostrict__trans,axiom,
    ! [Aa: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% order.strict_trans
tff(fact_1379_order_Ostrict__trans,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% order.strict_trans
tff(fact_1380_order_Ostrict__trans,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% order.strict_trans
tff(fact_1381_not__less__iff__gr__or__eq,axiom,
    ! [Xb: real,Y3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb)
        | ( Xb = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1382_not__less__iff__gr__or__eq,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb)
        | ( Xb = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1383_not__less__iff__gr__or__eq,axiom,
    ! [Xb: num,Y3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb)
        | ( Xb = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1384_not__less__iff__gr__or__eq,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb)
        | ( Xb = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1385_not__less__iff__gr__or__eq,axiom,
    ! [Xb: int,Y3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb)
        | ( Xb = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1386_dual__order_Ostrict__trans,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ( 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) ) ) ).

% dual_order.strict_trans
tff(fact_1387_dual__order_Ostrict__trans,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa) ) ) ).

% dual_order.strict_trans
tff(fact_1388_dual__order_Ostrict__trans,axiom,
    ! [B: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa) ) ) ).

% dual_order.strict_trans
tff(fact_1389_dual__order_Ostrict__trans,axiom,
    ! [B: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
     => ( 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) ) ) ).

% dual_order.strict_trans
tff(fact_1390_dual__order_Ostrict__trans,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => ( 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) ) ) ).

% dual_order.strict_trans
tff(fact_1391_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( Aa != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1392_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( Aa != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1393_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => ( Aa != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1394_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( Aa != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1395_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ( Aa != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1396_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ( Aa != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1397_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ( Aa != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1398_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
     => ( Aa != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1399_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
     => ( Aa != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1400_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => ( Aa != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1401_linorder__neqE,axiom,
    ! [Xb: real,Y3: real] :
      ( ( Xb != Y3 )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ) ).

% linorder_neqE
tff(fact_1402_linorder__neqE,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ( Xb != Y3 )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ) ).

% linorder_neqE
tff(fact_1403_linorder__neqE,axiom,
    ! [Xb: num,Y3: num] :
      ( ( Xb != Y3 )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb) ) ) ).

% linorder_neqE
tff(fact_1404_linorder__neqE,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ( Xb != Y3 )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ) ).

% linorder_neqE
tff(fact_1405_linorder__neqE,axiom,
    ! [Xb: int,Y3: int] :
      ( ( Xb != Y3 )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ) ).

% linorder_neqE
tff(fact_1406_order__less__asym,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ).

% order_less_asym
tff(fact_1407_order__less__asym,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ).

% order_less_asym
tff(fact_1408_order__less__asym,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb) ) ).

% order_less_asym
tff(fact_1409_order__less__asym,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ).

% order_less_asym
tff(fact_1410_order__less__asym,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ).

% order_less_asym
tff(fact_1411_linorder__neq__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( ( Xb != Y3 )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1412_linorder__neq__iff,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ( Xb != Y3 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1413_linorder__neq__iff,axiom,
    ! [Xb: num,Y3: num] :
      ( ( Xb != Y3 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1414_linorder__neq__iff,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ( Xb != Y3 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1415_linorder__neq__iff,axiom,
    ! [Xb: int,Y3: int] :
      ( ( Xb != Y3 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1416_order__less__asym_H,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa) ) ).

% order_less_asym'
tff(fact_1417_order__less__asym_H,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa) ) ).

% order_less_asym'
tff(fact_1418_order__less__asym_H,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa) ) ).

% order_less_asym'
tff(fact_1419_order__less__asym_H,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa) ) ).

% order_less_asym'
tff(fact_1420_order__less__asym_H,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa) ) ).

% order_less_asym'
tff(fact_1421_order__less__trans,axiom,
    ! [Xb: real,Y3: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1422_order__less__trans,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1423_order__less__trans,axiom,
    ! [Xb: num,Y3: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1424_order__less__trans,axiom,
    ! [Xb: nat,Y3: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1425_order__less__trans,axiom,
    ! [Xb: int,Y3: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1426_ord__eq__less__subst,axiom,
    ! [Aa: real,F: fun_real_real,B: real,C: real] :
      ( ( Aa = aa_real_real(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1427_ord__eq__less__subst,axiom,
    ! [Aa: rat,F: fun_real_rat,B: real,C: real] :
      ( ( Aa = aa_real_rat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_real_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1428_ord__eq__less__subst,axiom,
    ! [Aa: num,F: fun_real_num,B: real,C: real] :
      ( ( Aa = aa_real_num(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_real_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1429_ord__eq__less__subst,axiom,
    ! [Aa: nat,F: fun_real_nat,B: real,C: real] :
      ( ( Aa = aa_real_nat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_real_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1430_ord__eq__less__subst,axiom,
    ! [Aa: int,F: fun_real_int,B: real,C: real] :
      ( ( Aa = aa_real_int(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X2)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_real_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1431_ord__eq__less__subst,axiom,
    ! [Aa: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( Aa = aa_rat_real(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1432_ord__eq__less__subst,axiom,
    ! [Aa: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( Aa = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1433_ord__eq__less__subst,axiom,
    ! [Aa: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( Aa = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1434_ord__eq__less__subst,axiom,
    ! [Aa: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( Aa = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1435_ord__eq__less__subst,axiom,
    ! [Aa: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( Aa = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1436_ord__less__eq__subst,axiom,
    ! [Aa: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1437_ord__less__eq__subst,axiom,
    ! [Aa: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1438_ord__less__eq__subst,axiom,
    ! [Aa: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1439_ord__less__eq__subst,axiom,
    ! [Aa: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1440_ord__less__eq__subst,axiom,
    ! [Aa: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X2)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1441_ord__less__eq__subst,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1442_ord__less__eq__subst,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1443_ord__less__eq__subst,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1444_ord__less__eq__subst,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1445_ord__less__eq__subst,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1446_order__less__irrefl,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Xb) ).

% order_less_irrefl
tff(fact_1447_order__less__irrefl,axiom,
    ! [Xb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Xb) ).

% order_less_irrefl
tff(fact_1448_order__less__irrefl,axiom,
    ! [Xb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Xb) ).

% order_less_irrefl
tff(fact_1449_order__less__irrefl,axiom,
    ! [Xb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xb) ).

% order_less_irrefl
tff(fact_1450_order__less__irrefl,axiom,
    ! [Xb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xb) ).

% order_less_irrefl
tff(fact_1451_order__less__subst1,axiom,
    ! [Aa: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1452_order__less__subst1,axiom,
    ! [Aa: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1453_order__less__subst1,axiom,
    ! [Aa: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1454_order__less__subst1,axiom,
    ! [Aa: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_nat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1455_order__less__subst1,axiom,
    ! [Aa: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_int_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1456_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_real_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1457_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1458_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1459_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_nat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1460_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_int_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1461_order__less__subst2,axiom,
    ! [Aa: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1462_order__less__subst2,axiom,
    ! [Aa: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1463_order__less__subst2,axiom,
    ! [Aa: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1464_order__less__subst2,axiom,
    ! [Aa: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1465_order__less__subst2,axiom,
    ! [Aa: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X2)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1466_order__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1467_order__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1468_order__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1469_order__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1470_order__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_1471_order__less__not__sym,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ).

% order_less_not_sym
tff(fact_1472_order__less__not__sym,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ).

% order_less_not_sym
tff(fact_1473_order__less__not__sym,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb) ) ).

% order_less_not_sym
tff(fact_1474_order__less__not__sym,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ).

% order_less_not_sym
tff(fact_1475_order__less__not__sym,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ).

% order_less_not_sym
tff(fact_1476_order__less__imp__triv,axiom,
    ! [Xb: real,Y3: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1477_order__less__imp__triv,axiom,
    ! [Xb: rat,Y3: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1478_order__less__imp__triv,axiom,
    ! [Xb: num,Y3: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1479_order__less__imp__triv,axiom,
    ! [Xb: nat,Y3: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1480_order__less__imp__triv,axiom,
    ! [Xb: int,Y3: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1481_linorder__less__linear,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
      | ( Xb = Y3 )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ).

% linorder_less_linear
tff(fact_1482_linorder__less__linear,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
      | ( Xb = Y3 )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ).

% linorder_less_linear
tff(fact_1483_linorder__less__linear,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
      | ( Xb = Y3 )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb) ) ).

% linorder_less_linear
tff(fact_1484_linorder__less__linear,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
      | ( Xb = Y3 )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ).

% linorder_less_linear
tff(fact_1485_linorder__less__linear,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
      | ( Xb = Y3 )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ).

% linorder_less_linear
tff(fact_1486_order__less__imp__not__eq,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_1487_order__less__imp__not__eq,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_1488_order__less__imp__not__eq,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_1489_order__less__imp__not__eq,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_1490_order__less__imp__not__eq,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ( Xb != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_1491_order__less__imp__not__eq2,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ( Y3 != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1492_order__less__imp__not__eq2,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ( Y3 != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1493_order__less__imp__not__eq2,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ( Y3 != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1494_order__less__imp__not__eq2,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ( Y3 != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1495_order__less__imp__not__eq2,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ( Y3 != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1496_order__less__imp__not__less,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ).

% order_less_imp_not_less
tff(fact_1497_order__less__imp__not__less,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ).

% order_less_imp_not_less
tff(fact_1498_order__less__imp__not__less,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb) ) ).

% order_less_imp_not_less
tff(fact_1499_order__less__imp__not__less,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ).

% order_less_imp_not_less
tff(fact_1500_order__less__imp__not__less,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ).

% order_less_imp_not_less
tff(fact_1501_max_Oassoc,axiom,
    ! [Aa: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)) ).

% max.assoc
tff(fact_1502_max_Oassoc,axiom,
    ! [Aa: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)),C) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)) ).

% max.assoc
tff(fact_1503_max_Oassoc,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.assoc
tff(fact_1504_max_Oassoc,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] : ord_max_Code_integer(ord_max_Code_integer(Aa,B),C) = ord_max_Code_integer(Aa,ord_max_Code_integer(B,C)) ).

% max.assoc
tff(fact_1505_max_Ocommute,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),Aa) ).

% max.commute
tff(fact_1506_max_Ocommute,axiom,
    ! [Aa: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),Aa) ).

% max.commute
tff(fact_1507_max_Ocommute,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,B),Aa) ).

% max.commute
tff(fact_1508_max_Ocommute,axiom,
    ! [Aa: code_integer,B: code_integer] : ord_max_Code_integer(Aa,B) = ord_max_Code_integer(B,Aa) ).

% max.commute
tff(fact_1509_max_Oleft__commute,axiom,
    ! [B: nat,Aa: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),C)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)) ).

% max.left_commute
tff(fact_1510_max_Oleft__commute,axiom,
    ! [B: extended_enat,Aa: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),C)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)) ).

% max.left_commute
tff(fact_1511_max_Oleft__commute,axiom,
    ! [B: int,Aa: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),C)) = aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.left_commute
tff(fact_1512_max_Oleft__commute,axiom,
    ! [B: code_integer,Aa: code_integer,C: code_integer] : ord_max_Code_integer(B,ord_max_Code_integer(Aa,C)) = ord_max_Code_integer(Aa,ord_max_Code_integer(B,C)) ).

% max.left_commute
tff(fact_1513_order__le__imp__less__or__eq,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1514_order__le__imp__less__or__eq,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
     => ( aa_set_int_o(ord_less_set_int(Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1515_order__le__imp__less__or__eq,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1516_order__le__imp__less__or__eq,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1517_order__le__imp__less__or__eq,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1518_order__le__imp__less__or__eq,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1519_linorder__le__less__linear,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ).

% linorder_le_less_linear
tff(fact_1520_linorder__le__less__linear,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ).

% linorder_le_less_linear
tff(fact_1521_linorder__le__less__linear,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb) ) ).

% linorder_le_less_linear
tff(fact_1522_linorder__le__less__linear,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ).

% linorder_le_less_linear
tff(fact_1523_linorder__le__less__linear,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ).

% linorder_le_less_linear
tff(fact_1524_order__less__le__subst2,axiom,
    ! [Aa: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1525_order__less__le__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1526_order__less__le__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1527_order__less__le__subst2,axiom,
    ! [Aa: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1528_order__less__le__subst2,axiom,
    ! [Aa: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1529_order__less__le__subst2,axiom,
    ! [Aa: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B)),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1530_order__less__le__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1531_order__less__le__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1532_order__less__le__subst2,axiom,
    ! [Aa: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1533_order__less__le__subst2,axiom,
    ! [Aa: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1534_order__less__le__subst1,axiom,
    ! [Aa: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1535_order__less__le__subst1,axiom,
    ! [Aa: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1536_order__less__le__subst1,axiom,
    ! [Aa: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_rat_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1537_order__less__le__subst1,axiom,
    ! [Aa: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1538_order__less__le__subst1,axiom,
    ! [Aa: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1539_order__less__le__subst1,axiom,
    ! [Aa: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1540_order__less__le__subst1,axiom,
    ! [Aa: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1541_order__less__le__subst1,axiom,
    ! [Aa: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_num_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1542_order__less__le__subst1,axiom,
    ! [Aa: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_num_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1543_order__less__le__subst1,axiom,
    ! [Aa: int,F: fun_num_int,B: num,C: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_num_int(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_num_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1544_order__le__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1545_order__le__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1546_order__le__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1547_order__le__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1548_order__le__less__subst2,axiom,
    ! [Aa: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1549_order__le__less__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1550_order__le__less__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1551_order__le__less__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1552_order__le__less__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1553_order__le__less__subst2,axiom,
    ! [Aa: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B)),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1554_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1555_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1556_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1557_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_nat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1558_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_int_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1559_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_real_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1560_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1561_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1562_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_nat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1563_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_int_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1564_order__less__le__trans,axiom,
    ! [Xb: real,Y3: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1565_order__less__le__trans,axiom,
    ! [Xb: set_int,Y3: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Y3)
     => ( aa_set_int_o(ord_less_eq_set_int(Y3),Z)
       => aa_set_int_o(ord_less_set_int(Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1566_order__less__le__trans,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1567_order__less__le__trans,axiom,
    ! [Xb: num,Y3: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1568_order__less__le__trans,axiom,
    ! [Xb: nat,Y3: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1569_order__less__le__trans,axiom,
    ! [Xb: int,Y3: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1570_order__le__less__trans,axiom,
    ! [Xb: real,Y3: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1571_order__le__less__trans,axiom,
    ! [Xb: set_int,Y3: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
     => ( aa_set_int_o(ord_less_set_int(Y3),Z)
       => aa_set_int_o(ord_less_set_int(Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1572_order__le__less__trans,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1573_order__le__less__trans,axiom,
    ! [Xb: num,Y3: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1574_order__le__less__trans,axiom,
    ! [Xb: nat,Y3: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1575_order__le__less__trans,axiom,
    ! [Xb: int,Y3: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1576_order__neq__le__trans,axiom,
    ! [Aa: real,B: real] :
      ( ( Aa != B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ) ).

% order_neq_le_trans
tff(fact_1577_order__neq__le__trans,axiom,
    ! [Aa: set_int,B: set_int] :
      ( ( Aa != B )
     => ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
       => aa_set_int_o(ord_less_set_int(Aa),B) ) ) ).

% order_neq_le_trans
tff(fact_1578_order__neq__le__trans,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ) ).

% order_neq_le_trans
tff(fact_1579_order__neq__le__trans,axiom,
    ! [Aa: num,B: num] :
      ( ( Aa != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B) ) ) ).

% order_neq_le_trans
tff(fact_1580_order__neq__le__trans,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B) ) ) ).

% order_neq_le_trans
tff(fact_1581_order__neq__le__trans,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B) ) ) ).

% order_neq_le_trans
tff(fact_1582_order__le__neq__trans,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => ( ( Aa != B )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ) ).

% order_le_neq_trans
tff(fact_1583_order__le__neq__trans,axiom,
    ! [Aa: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
     => ( ( Aa != B )
       => aa_set_int_o(ord_less_set_int(Aa),B) ) ) ).

% order_le_neq_trans
tff(fact_1584_order__le__neq__trans,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( ( Aa != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ) ).

% order_le_neq_trans
tff(fact_1585_order__le__neq__trans,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( ( Aa != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B) ) ) ).

% order_le_neq_trans
tff(fact_1586_order__le__neq__trans,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( ( Aa != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B) ) ) ).

% order_le_neq_trans
tff(fact_1587_order__le__neq__trans,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => ( ( Aa != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B) ) ) ).

% order_le_neq_trans
tff(fact_1588_order__less__imp__le,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ).

% order_less_imp_le
tff(fact_1589_order__less__imp__le,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Y3)
     => aa_set_int_o(ord_less_eq_set_int(Xb),Y3) ) ).

% order_less_imp_le
tff(fact_1590_order__less__imp__le,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3) ) ).

% order_less_imp_le
tff(fact_1591_order__less__imp__le,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3) ) ).

% order_less_imp_le
tff(fact_1592_order__less__imp__le,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3) ) ).

% order_less_imp_le
tff(fact_1593_order__less__imp__le,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3) ) ).

% order_less_imp_le
tff(fact_1594_linorder__not__less,axiom,
    ! [Xb: real,Y3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb) ) ).

% linorder_not_less
tff(fact_1595_linorder__not__less,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb) ) ).

% linorder_not_less
tff(fact_1596_linorder__not__less,axiom,
    ! [Xb: num,Y3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb) ) ).

% linorder_not_less
tff(fact_1597_linorder__not__less,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb) ) ).

% linorder_not_less
tff(fact_1598_linorder__not__less,axiom,
    ! [Xb: int,Y3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb) ) ).

% linorder_not_less
tff(fact_1599_linorder__not__le,axiom,
    ! [Xb: real,Y3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ).

% linorder_not_le
tff(fact_1600_linorder__not__le,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ) ).

% linorder_not_le
tff(fact_1601_linorder__not__le,axiom,
    ! [Xb: num,Y3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y3),Xb) ) ).

% linorder_not_le
tff(fact_1602_linorder__not__le,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),Xb) ) ).

% linorder_not_le
tff(fact_1603_linorder__not__le,axiom,
    ! [Xb: int,Y3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ) ).

% linorder_not_le
tff(fact_1604_order__less__le,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
        & ( Xb != Y3 ) ) ) ).

% order_less_le
tff(fact_1605_order__less__le,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Y3)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
        & ( Xb != Y3 ) ) ) ).

% order_less_le
tff(fact_1606_order__less__le,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
        & ( Xb != Y3 ) ) ) ).

% order_less_le
tff(fact_1607_order__less__le,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
        & ( Xb != Y3 ) ) ) ).

% order_less_le
tff(fact_1608_order__less__le,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
        & ( Xb != Y3 ) ) ) ).

% order_less_le
tff(fact_1609_order__less__le,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
        & ( Xb != Y3 ) ) ) ).

% order_less_le
tff(fact_1610_order__le__less,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_less
tff(fact_1611_order__le__less,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
    <=> ( aa_set_int_o(ord_less_set_int(Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_less
tff(fact_1612_order__le__less,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_less
tff(fact_1613_order__le__less,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_less
tff(fact_1614_order__le__less,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_less
tff(fact_1615_order__le__less,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% order_le_less
tff(fact_1616_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_1617_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),Aa)
     => aa_set_int_o(ord_less_eq_set_int(B),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_1618_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_1619_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_1620_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_1621_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_1622_order_Ostrict__implies__order,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ).

% order.strict_implies_order
tff(fact_1623_order_Ostrict__implies__order,axiom,
    ! [Aa: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(Aa),B)
     => aa_set_int_o(ord_less_eq_set_int(Aa),B) ) ).

% order.strict_implies_order
tff(fact_1624_order_Ostrict__implies__order,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ).

% order.strict_implies_order
tff(fact_1625_order_Ostrict__implies__order,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B) ) ).

% order.strict_implies_order
tff(fact_1626_order_Ostrict__implies__order,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B) ) ).

% order.strict_implies_order
tff(fact_1627_order_Ostrict__implies__order,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B) ) ).

% order.strict_implies_order
tff(fact_1628_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1629_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),Aa)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),Aa)
        & ~ aa_set_int_o(ord_less_eq_set_int(Aa),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1630_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1631_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1632_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1633_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1634_dual__order_Ostrict__trans2,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_1635_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,Aa: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),Aa)
     => ( aa_set_int_o(ord_less_eq_set_int(C),B)
       => aa_set_int_o(ord_less_set_int(C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_1636_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_1637_dual__order_Ostrict__trans2,axiom,
    ! [B: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_1638_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_1639_dual__order_Ostrict__trans2,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_1640_dual__order_Ostrict__trans1,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
     => ( 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) ) ) ).

% dual_order.strict_trans1
tff(fact_1641_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,Aa: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),Aa)
     => ( aa_set_int_o(ord_less_set_int(C),B)
       => aa_set_int_o(ord_less_set_int(C),Aa) ) ) ).

% dual_order.strict_trans1
tff(fact_1642_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa) ) ) ).

% dual_order.strict_trans1
tff(fact_1643_dual__order_Ostrict__trans1,axiom,
    ! [B: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa) ) ) ).

% dual_order.strict_trans1
tff(fact_1644_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
     => ( 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) ) ) ).

% dual_order.strict_trans1
tff(fact_1645_dual__order_Ostrict__trans1,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( 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) ) ) ).

% dual_order.strict_trans1
tff(fact_1646_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
        & ( Aa != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1647_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),Aa)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),Aa)
        & ( Aa != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1648_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
        & ( Aa != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1649_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
        & ( Aa != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1650_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
        & ( Aa != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1651_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
        & ( Aa != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1652_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
        | ( Aa = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1653_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),Aa)
    <=> ( aa_set_int_o(ord_less_set_int(B),Aa)
        | ( Aa = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1654_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
        | ( Aa = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1655_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
        | ( Aa = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1656_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
        | ( Aa = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1657_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
        | ( Aa = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1658_dense__le__bounded,axiom,
    ! [Xb: real,Y3: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Z) ) ) ).

% dense_le_bounded
tff(fact_1659_dense__le__bounded,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Z) ) ) ).

% dense_le_bounded
tff(fact_1660_dense__ge__bounded,axiom,
    ! [Z: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Xb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),W2) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Z) ) ) ).

% dense_ge_bounded
tff(fact_1661_dense__ge__bounded,axiom,
    ! [Z: rat,Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Xb)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Z) ) ) ).

% dense_ge_bounded
tff(fact_1662_order_Ostrict__iff__not,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa) ) ) ).

% order.strict_iff_not
tff(fact_1663_order_Ostrict__iff__not,axiom,
    ! [Aa: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(Aa),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
        & ~ aa_set_int_o(ord_less_eq_set_int(B),Aa) ) ) ).

% order.strict_iff_not
tff(fact_1664_order_Ostrict__iff__not,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ) ).

% order.strict_iff_not
tff(fact_1665_order_Ostrict__iff__not,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa) ) ) ).

% order.strict_iff_not
tff(fact_1666_order_Ostrict__iff__not,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa) ) ) ).

% order.strict_iff_not
tff(fact_1667_order_Ostrict__iff__not,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa) ) ) ).

% order.strict_iff_not
tff(fact_1668_order_Ostrict__trans2,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( 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,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_1669_order_Ostrict__trans2,axiom,
    ! [Aa: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_set_int(Aa),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => aa_set_int_o(ord_less_set_int(Aa),C) ) ) ).

% order.strict_trans2
tff(fact_1670_order_Ostrict__trans2,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_1671_order_Ostrict__trans2,axiom,
    ! [Aa: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_1672_order_Ostrict__trans2,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_1673_order_Ostrict__trans2,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_1674_order_Ostrict__trans1,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_1675_order_Ostrict__trans1,axiom,
    ! [Aa: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
     => ( aa_set_int_o(ord_less_set_int(B),C)
       => aa_set_int_o(ord_less_set_int(Aa),C) ) ) ).

% order.strict_trans1
tff(fact_1676_order_Ostrict__trans1,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_1677_order_Ostrict__trans1,axiom,
    ! [Aa: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_1678_order_Ostrict__trans1,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_1679_order_Ostrict__trans1,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_1680_order_Ostrict__iff__order,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
        & ( Aa != B ) ) ) ).

% order.strict_iff_order
tff(fact_1681_order_Ostrict__iff__order,axiom,
    ! [Aa: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(Aa),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
        & ( Aa != B ) ) ) ).

% order.strict_iff_order
tff(fact_1682_order_Ostrict__iff__order,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
        & ( Aa != B ) ) ) ).

% order.strict_iff_order
tff(fact_1683_order_Ostrict__iff__order,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
        & ( Aa != B ) ) ) ).

% order.strict_iff_order
tff(fact_1684_order_Ostrict__iff__order,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
        & ( Aa != B ) ) ) ).

% order.strict_iff_order
tff(fact_1685_order_Ostrict__iff__order,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
        & ( Aa != B ) ) ) ).

% order.strict_iff_order
tff(fact_1686_order_Oorder__iff__strict,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
        | ( Aa = B ) ) ) ).

% order.order_iff_strict
tff(fact_1687_order_Oorder__iff__strict,axiom,
    ! [Aa: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),B)
    <=> ( aa_set_int_o(ord_less_set_int(Aa),B)
        | ( Aa = B ) ) ) ).

% order.order_iff_strict
tff(fact_1688_order_Oorder__iff__strict,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
        | ( Aa = B ) ) ) ).

% order.order_iff_strict
tff(fact_1689_order_Oorder__iff__strict,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
        | ( Aa = B ) ) ) ).

% order.order_iff_strict
tff(fact_1690_order_Oorder__iff__strict,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
        | ( Aa = B ) ) ) ).

% order.order_iff_strict
tff(fact_1691_order_Oorder__iff__strict,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
        | ( Aa = B ) ) ) ).

% order.order_iff_strict
tff(fact_1692_not__le__imp__less,axiom,
    ! [Y3: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ).

% not_le_imp_less
tff(fact_1693_not__le__imp__less,axiom,
    ! [Y3: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3) ) ).

% not_le_imp_less
tff(fact_1694_not__le__imp__less,axiom,
    ! [Y3: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3) ) ).

% not_le_imp_less
tff(fact_1695_not__le__imp__less,axiom,
    ! [Y3: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3) ) ).

% not_le_imp_less
tff(fact_1696_not__le__imp__less,axiom,
    ! [Y3: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3) ) ).

% not_le_imp_less
tff(fact_1697_less__le__not__le,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb) ) ) ).

% less_le_not_le
tff(fact_1698_less__le__not__le,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Y3)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
        & ~ aa_set_int_o(ord_less_eq_set_int(Y3),Xb) ) ) ).

% less_le_not_le
tff(fact_1699_less__le__not__le,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb) ) ) ).

% less_le_not_le
tff(fact_1700_less__le__not__le,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb) ) ) ).

% less_le_not_le
tff(fact_1701_less__le__not__le,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb) ) ) ).

% less_le_not_le
tff(fact_1702_less__le__not__le,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb) ) ) ).

% less_le_not_le
tff(fact_1703_dense__le,axiom,
    ! [Y3: real,Z: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Z) ) ).

% dense_le
tff(fact_1704_dense__le,axiom,
    ! [Y3: rat,Z: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Z) ) ).

% dense_le
tff(fact_1705_dense__ge,axiom,
    ! [Z: real,Y3: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),X2) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Z) ) ).

% dense_ge
tff(fact_1706_dense__ge,axiom,
    ! [Z: rat,Y3: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),X2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Z) ) ).

% dense_ge
tff(fact_1707_antisym__conv2,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv2
tff(fact_1708_antisym__conv2,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
     => ( ~ aa_set_int_o(ord_less_set_int(Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv2
tff(fact_1709_antisym__conv2,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv2
tff(fact_1710_antisym__conv2,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv2
tff(fact_1711_antisym__conv2,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv2
tff(fact_1712_antisym__conv2,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv2
tff(fact_1713_antisym__conv1,axiom,
    ! [Xb: real,Y3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv1
tff(fact_1714_antisym__conv1,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(Xb),Y3)
     => ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv1
tff(fact_1715_antisym__conv1,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv1
tff(fact_1716_antisym__conv1,axiom,
    ! [Xb: num,Y3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv1
tff(fact_1717_antisym__conv1,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv1
tff(fact_1718_antisym__conv1,axiom,
    ! [Xb: int,Y3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
      <=> ( Xb = Y3 ) ) ) ).

% antisym_conv1
tff(fact_1719_nless__le,axiom,
    ! [Aa: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
        | ( Aa = B ) ) ) ).

% nless_le
tff(fact_1720_nless__le,axiom,
    ! [Aa: set_int,B: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(Aa),B)
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(Aa),B)
        | ( Aa = B ) ) ) ).

% nless_le
tff(fact_1721_nless__le,axiom,
    ! [Aa: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
        | ( Aa = B ) ) ) ).

% nless_le
tff(fact_1722_nless__le,axiom,
    ! [Aa: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
        | ( Aa = B ) ) ) ).

% nless_le
tff(fact_1723_nless__le,axiom,
    ! [Aa: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
        | ( Aa = B ) ) ) ).

% nless_le
tff(fact_1724_nless__le,axiom,
    ! [Aa: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
        | ( Aa = B ) ) ) ).

% nless_le
tff(fact_1725_leI,axiom,
    ! [Xb: real,Y3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb) ) ).

% leI
tff(fact_1726_leI,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb) ) ).

% leI
tff(fact_1727_leI,axiom,
    ! [Xb: num,Y3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb) ) ).

% leI
tff(fact_1728_leI,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb) ) ).

% leI
tff(fact_1729_leI,axiom,
    ! [Xb: int,Y3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb) ) ).

% leI
tff(fact_1730_leD,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ).

% leD
tff(fact_1731_leD,axiom,
    ! [Y3: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y3),Xb)
     => ~ aa_set_int_o(ord_less_set_int(Xb),Y3) ) ).

% leD
tff(fact_1732_leD,axiom,
    ! [Y3: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3) ) ).

% leD
tff(fact_1733_leD,axiom,
    ! [Y3: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y3) ) ).

% leD
tff(fact_1734_leD,axiom,
    ! [Y3: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3) ) ).

% leD
tff(fact_1735_leD,axiom,
    ! [Y3: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3) ) ).

% leD
tff(fact_1736_bot_Oextremum__uniqueI,axiom,
    ! [Aa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Aa),bot_bot_set_nat)
     => ( Aa = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1737_bot_Oextremum__uniqueI,axiom,
    ! [Aa: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(Aa),bot_bot_set_real)
     => ( Aa = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_1738_bot_Oextremum__uniqueI,axiom,
    ! [Aa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),bot_bot_set_int)
     => ( Aa = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_1739_bot_Oextremum__uniqueI,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),bot_bot_nat)
     => ( Aa = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1740_bot_Oextremum__unique,axiom,
    ! [Aa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Aa),bot_bot_set_nat)
    <=> ( Aa = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_1741_bot_Oextremum__unique,axiom,
    ! [Aa: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(Aa),bot_bot_set_real)
    <=> ( Aa = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_1742_bot_Oextremum__unique,axiom,
    ! [Aa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),bot_bot_set_int)
    <=> ( Aa = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_1743_bot_Oextremum__unique,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),bot_bot_nat)
    <=> ( Aa = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_1744_bot_Oextremum,axiom,
    ! [Aa: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),Aa) ).

% bot.extremum
tff(fact_1745_bot_Oextremum,axiom,
    ! [Aa: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),Aa) ).

% bot.extremum
tff(fact_1746_bot_Oextremum,axiom,
    ! [Aa: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),Aa) ).

% bot.extremum
tff(fact_1747_bot_Oextremum,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),Aa) ).

% bot.extremum
tff(fact_1748_bot_Oextremum__strict,axiom,
    ! [Aa: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(Aa),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_1749_bot_Oextremum__strict,axiom,
    ! [Aa: set_int] : ~ aa_set_int_o(ord_less_set_int(Aa),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_1750_bot_Oextremum__strict,axiom,
    ! [Aa: set_real] : ~ aa_set_real_o(ord_less_set_real(Aa),bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_1751_bot_Oextremum__strict,axiom,
    ! [Aa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_1752_bot_Onot__eq__extremum,axiom,
    ! [Aa: set_nat] :
      ( ( Aa != bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_set_nat(bot_bot_set_nat),Aa) ) ).

% bot.not_eq_extremum
tff(fact_1753_bot_Onot__eq__extremum,axiom,
    ! [Aa: set_int] :
      ( ( Aa != bot_bot_set_int )
    <=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),Aa) ) ).

% bot.not_eq_extremum
tff(fact_1754_bot_Onot__eq__extremum,axiom,
    ! [Aa: set_real] :
      ( ( Aa != bot_bot_set_real )
    <=> aa_set_real_o(ord_less_set_real(bot_bot_set_real),Aa) ) ).

% bot.not_eq_extremum
tff(fact_1755_bot_Onot__eq__extremum,axiom,
    ! [Aa: nat] :
      ( ( Aa != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),Aa) ) ).

% bot.not_eq_extremum
tff(fact_1756_max__absorb2,axiom,
    ! [Xb: extended_enat,Y3: extended_enat] :
      ( ord_le2932123472753598470d_enat(Xb,Y3)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_1757_max__absorb2,axiom,
    ! [Xb: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Y3)
     => ( ord_max_Code_integer(Xb,Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_1758_max__absorb2,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
     => ( ord_max_set_int(Xb,Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_1759_max__absorb2,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
     => ( aa_rat_rat(ord_max_rat(Xb),Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_1760_max__absorb2,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_1761_max__absorb2,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_1762_max__absorb2,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_1763_max__absorb1,axiom,
    ! [Y3: extended_enat,Xb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Y3,Xb)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y3) = Xb ) ) ).

% max_absorb1
tff(fact_1764_max__absorb1,axiom,
    ! [Y3: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y3),Xb)
     => ( ord_max_Code_integer(Xb,Y3) = Xb ) ) ).

% max_absorb1
tff(fact_1765_max__absorb1,axiom,
    ! [Y3: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y3),Xb)
     => ( ord_max_set_int(Xb,Y3) = Xb ) ) ).

% max_absorb1
tff(fact_1766_max__absorb1,axiom,
    ! [Y3: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb)
     => ( aa_rat_rat(ord_max_rat(Xb),Y3) = Xb ) ) ).

% max_absorb1
tff(fact_1767_max__absorb1,axiom,
    ! [Y3: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xb)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y3) = Xb ) ) ).

% max_absorb1
tff(fact_1768_max__absorb1,axiom,
    ! [Y3: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y3) = Xb ) ) ).

% max_absorb1
tff(fact_1769_max__absorb1,axiom,
    ! [Y3: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xb)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y3) = Xb ) ) ).

% max_absorb1
tff(fact_1770_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,Aa: extended_enat] :
      ( ord_le2932123472753598470d_enat(C,B)
     => ord_le2932123472753598470d_enat(C,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)) ) ).

% max.coboundedI2
tff(fact_1771_max_OcoboundedI2,axiom,
    ! [C: code_integer,B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),ord_max_Code_integer(Aa,B)) ) ).

% max.coboundedI2
tff(fact_1772_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(Aa),B)) ) ).

% max.coboundedI2
tff(fact_1773_max_OcoboundedI2,axiom,
    ! [C: num,B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B)) ) ).

% max.coboundedI2
tff(fact_1774_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,Aa: 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,Aa),B)) ) ).

% max.coboundedI2
tff(fact_1775_max_OcoboundedI2,axiom,
    ! [C: int,B: int,Aa: 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,Aa),B)) ) ).

% max.coboundedI2
tff(fact_1776_max_OcoboundedI1,axiom,
    ! [C: extended_enat,Aa: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(C,Aa)
     => ord_le2932123472753598470d_enat(C,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)) ) ).

% max.coboundedI1
tff(fact_1777_max_OcoboundedI1,axiom,
    ! [C: code_integer,Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),ord_max_Code_integer(Aa,B)) ) ).

% max.coboundedI1
tff(fact_1778_max_OcoboundedI1,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(Aa),B)) ) ).

% max.coboundedI1
tff(fact_1779_max_OcoboundedI1,axiom,
    ! [C: num,Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B)) ) ).

% max.coboundedI1
tff(fact_1780_max_OcoboundedI1,axiom,
    ! [C: nat,Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B)) ) ).

% max.coboundedI1
tff(fact_1781_max_OcoboundedI1,axiom,
    ! [C: int,Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B)) ) ).

% max.coboundedI1
tff(fact_1782_max_Oabsorb__iff2,axiom,
    ! [Aa: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(Aa,B)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1783_max_Oabsorb__iff2,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),B)
    <=> ( ord_max_Code_integer(Aa,B) = B ) ) ).

% max.absorb_iff2
tff(fact_1784_max_Oabsorb__iff2,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
    <=> ( aa_rat_rat(ord_max_rat(Aa),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1785_max_Oabsorb__iff2,axiom,
    ! [Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1786_max_Oabsorb__iff2,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1787_max_Oabsorb__iff2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1788_max_Oabsorb__iff1,axiom,
    ! [B: extended_enat,Aa: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,Aa)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) = Aa ) ) ).

% max.absorb_iff1
tff(fact_1789_max_Oabsorb__iff1,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Aa)
    <=> ( ord_max_Code_integer(Aa,B) = Aa ) ) ).

% max.absorb_iff1
tff(fact_1790_max_Oabsorb__iff1,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
    <=> ( aa_rat_rat(ord_max_rat(Aa),B) = Aa ) ) ).

% max.absorb_iff1
tff(fact_1791_max_Oabsorb__iff1,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) = Aa ) ) ).

% max.absorb_iff1
tff(fact_1792_max_Oabsorb__iff1,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) = Aa ) ) ).

% max.absorb_iff1
tff(fact_1793_max_Oabsorb__iff1,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) = Aa ) ) ).

% max.absorb_iff1
tff(fact_1794_le__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y3: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y3))
    <=> ( ord_le2932123472753598470d_enat(Z,Xb)
        | ord_le2932123472753598470d_enat(Z,Y3) ) ) ).

% le_max_iff_disj
tff(fact_1795_le__max__iff__disj,axiom,
    ! [Z: code_integer,Xb: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),ord_max_Code_integer(Xb,Y3))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Xb)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Y3) ) ) ).

% le_max_iff_disj
tff(fact_1796_le__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(Xb),Y3))
    <=> ( 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),Y3) ) ) ).

% le_max_iff_disj
tff(fact_1797_le__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y3))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y3) ) ) ).

% le_max_iff_disj
tff(fact_1798_le__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y3: 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),Y3))
    <=> ( 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),Y3) ) ) ).

% le_max_iff_disj
tff(fact_1799_le__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y3: 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),Y3))
    <=> ( 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),Y3) ) ) ).

% le_max_iff_disj
tff(fact_1800_max_Ocobounded2,axiom,
    ! [B: extended_enat,Aa: extended_enat] : ord_le2932123472753598470d_enat(B,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)) ).

% max.cobounded2
tff(fact_1801_max_Ocobounded2,axiom,
    ! [B: code_integer,Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),ord_max_Code_integer(Aa,B)) ).

% max.cobounded2
tff(fact_1802_max_Ocobounded2,axiom,
    ! [B: rat,Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(Aa),B)) ).

% max.cobounded2
tff(fact_1803_max_Ocobounded2,axiom,
    ! [B: num,Aa: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B)) ).

% max.cobounded2
tff(fact_1804_max_Ocobounded2,axiom,
    ! [B: nat,Aa: 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,Aa),B)) ).

% max.cobounded2
tff(fact_1805_max_Ocobounded2,axiom,
    ! [B: int,Aa: 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,Aa),B)) ).

% max.cobounded2
tff(fact_1806_max_Ocobounded1,axiom,
    ! [Aa: extended_enat,B: extended_enat] : ord_le2932123472753598470d_enat(Aa,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)) ).

% max.cobounded1
tff(fact_1807_max_Ocobounded1,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),ord_max_Code_integer(Aa,B)) ).

% max.cobounded1
tff(fact_1808_max_Ocobounded1,axiom,
    ! [Aa: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(ord_max_rat(Aa),B)) ).

% max.cobounded1
tff(fact_1809_max_Ocobounded1,axiom,
    ! [Aa: num,B: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B)) ).

% max.cobounded1
tff(fact_1810_max_Ocobounded1,axiom,
    ! [Aa: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B)) ).

% max.cobounded1
tff(fact_1811_max_Ocobounded1,axiom,
    ! [Aa: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B)) ).

% max.cobounded1
tff(fact_1812_max_Oorder__iff,axiom,
    ! [B: extended_enat,Aa: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,Aa)
    <=> ( Aa = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) ) ) ).

% max.order_iff
tff(fact_1813_max_Oorder__iff,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Aa)
    <=> ( Aa = ord_max_Code_integer(Aa,B) ) ) ).

% max.order_iff
tff(fact_1814_max_Oorder__iff,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
    <=> ( Aa = aa_rat_rat(ord_max_rat(Aa),B) ) ) ).

% max.order_iff
tff(fact_1815_max_Oorder__iff,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
    <=> ( Aa = aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) ) ) ).

% max.order_iff
tff(fact_1816_max_Oorder__iff,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
    <=> ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) ) ) ).

% max.order_iff
tff(fact_1817_max_Oorder__iff,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
    <=> ( Aa = aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) ) ) ).

% max.order_iff
tff(fact_1818_max_OboundedI,axiom,
    ! [B: extended_enat,Aa: extended_enat,C: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,Aa)
     => ( ord_le2932123472753598470d_enat(C,Aa)
       => ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C),Aa) ) ) ).

% max.boundedI
tff(fact_1819_max_OboundedI,axiom,
    ! [B: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),Aa) ) ) ).

% max.boundedI
tff(fact_1820_max_OboundedI,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),Aa) ) ) ).

% max.boundedI
tff(fact_1821_max_OboundedI,axiom,
    ! [B: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),Aa) ) ) ).

% max.boundedI
tff(fact_1822_max_OboundedI,axiom,
    ! [B: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
       => 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)),Aa) ) ) ).

% max.boundedI
tff(fact_1823_max_OboundedI,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
       => 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)),Aa) ) ) ).

% max.boundedI
tff(fact_1824_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,Aa: extended_enat] :
      ( ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C),Aa)
     => ~ ( ord_le2932123472753598470d_enat(B,Aa)
         => ~ ord_le2932123472753598470d_enat(C,Aa) ) ) ).

% max.boundedE
tff(fact_1825_max_OboundedE,axiom,
    ! [B: code_integer,C: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),Aa)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Aa)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa) ) ) ).

% max.boundedE
tff(fact_1826_max_OboundedE,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),Aa)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa) ) ) ).

% max.boundedE
tff(fact_1827_max_OboundedE,axiom,
    ! [B: num,C: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),Aa)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa) ) ) ).

% max.boundedE
tff(fact_1828_max_OboundedE,axiom,
    ! [B: nat,C: nat,Aa: 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)),Aa)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa) ) ) ).

% max.boundedE
tff(fact_1829_max_OboundedE,axiom,
    ! [B: int,C: int,Aa: 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)),Aa)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa) ) ) ).

% max.boundedE
tff(fact_1830_max_OorderI,axiom,
    ! [Aa: extended_enat,B: extended_enat] :
      ( ( Aa = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) )
     => ord_le2932123472753598470d_enat(B,Aa) ) ).

% max.orderI
tff(fact_1831_max_OorderI,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( Aa = ord_max_Code_integer(Aa,B) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Aa) ) ).

% max.orderI
tff(fact_1832_max_OorderI,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa = aa_rat_rat(ord_max_rat(Aa),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ).

% max.orderI
tff(fact_1833_max_OorderI,axiom,
    ! [Aa: num,B: num] :
      ( ( Aa = aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa) ) ).

% max.orderI
tff(fact_1834_max_OorderI,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa) ) ).

% max.orderI
tff(fact_1835_max_OorderI,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa) ) ).

% max.orderI
tff(fact_1836_max_OorderE,axiom,
    ! [B: extended_enat,Aa: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,Aa)
     => ( Aa = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) ) ) ).

% max.orderE
tff(fact_1837_max_OorderE,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),Aa)
     => ( Aa = ord_max_Code_integer(Aa,B) ) ) ).

% max.orderE
tff(fact_1838_max_OorderE,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( Aa = aa_rat_rat(ord_max_rat(Aa),B) ) ) ).

% max.orderE
tff(fact_1839_max_OorderE,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa)
     => ( Aa = aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) ) ) ).

% max.orderE
tff(fact_1840_max_OorderE,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa)
     => ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) ) ) ).

% max.orderE
tff(fact_1841_max_OorderE,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( Aa = aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) ) ) ).

% max.orderE
tff(fact_1842_max_Omono,axiom,
    ! [C: extended_enat,Aa: extended_enat,D: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(C,Aa)
     => ( ord_le2932123472753598470d_enat(D,B)
       => ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,C),D),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)) ) ) ).

% max.mono
tff(fact_1843_max_Omono,axiom,
    ! [C: code_integer,Aa: code_integer,D: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(C,D)),ord_max_Code_integer(Aa,B)) ) ) ).

% max.mono
tff(fact_1844_max_Omono,axiom,
    ! [C: rat,Aa: rat,D: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C),D)),aa_rat_rat(ord_max_rat(Aa),B)) ) ) ).

% max.mono
tff(fact_1845_max_Omono,axiom,
    ! [C: num,Aa: num,D: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B)) ) ) ).

% max.mono
tff(fact_1846_max_Omono,axiom,
    ! [C: nat,Aa: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
     => ( 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,Aa),B)) ) ) ).

% max.mono
tff(fact_1847_max_Omono,axiom,
    ! [C: int,Aa: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
     => ( 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,Aa),B)) ) ) ).

% max.mono
tff(fact_1848_max__def,axiom,
    ! [Aa: extended_enat,B: extended_enat] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) = $ite(ord_le2932123472753598470d_enat(Aa,B),B,Aa) ).

% max_def
tff(fact_1849_max__def,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ord_max_Code_integer(Aa,B) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),B),B,Aa) ).

% max_def
tff(fact_1850_max__def,axiom,
    ! [Aa: set_int,B: set_int] :
      ord_max_set_int(Aa,B) = $ite(aa_set_int_o(ord_less_eq_set_int(Aa),B),B,Aa) ).

% max_def
tff(fact_1851_max__def,axiom,
    ! [Aa: rat,B: rat] :
      aa_rat_rat(ord_max_rat(Aa),B) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B),B,Aa) ).

% max_def
tff(fact_1852_max__def,axiom,
    ! [Aa: num,B: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B),B,Aa) ).

% max_def
tff(fact_1853_max__def,axiom,
    ! [Aa: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B),B,Aa) ).

% max_def
tff(fact_1854_max__def,axiom,
    ! [Aa: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B),B,Aa) ).

% max_def
tff(fact_1855_less__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y3: extended_enat] :
      ( ord_le72135733267957522d_enat(Z,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y3))
    <=> ( ord_le72135733267957522d_enat(Z,Xb)
        | ord_le72135733267957522d_enat(Z,Y3) ) ) ).

% less_max_iff_disj
tff(fact_1856_less__max__iff__disj,axiom,
    ! [Z: code_integer,Xb: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),ord_max_Code_integer(Xb,Y3))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Xb)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Y3) ) ) ).

% less_max_iff_disj
tff(fact_1857_less__max__iff__disj,axiom,
    ! [Z: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(ord_max_real(Xb),Y3))
    <=> ( 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),Y3) ) ) ).

% less_max_iff_disj
tff(fact_1858_less__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(ord_max_rat(Xb),Y3))
    <=> ( 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),Y3) ) ) ).

% less_max_iff_disj
tff(fact_1859_less__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y3))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Xb)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Y3) ) ) ).

% less_max_iff_disj
tff(fact_1860_less__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y3: 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),Y3))
    <=> ( 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),Y3) ) ) ).

% less_max_iff_disj
tff(fact_1861_less__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y3: 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),Y3))
    <=> ( 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),Y3) ) ) ).

% less_max_iff_disj
tff(fact_1862_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,Aa: extended_enat] :
      ( ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C),Aa)
     => ~ ( ord_le72135733267957522d_enat(B,Aa)
         => ~ ord_le72135733267957522d_enat(C,Aa) ) ) ).

% max.strict_boundedE
tff(fact_1863_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ord_max_Code_integer(B,C)),Aa)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),Aa)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_1864_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(B),C)),Aa)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_1865_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(B),C)),Aa)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_1866_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),Aa)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_1867_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,Aa: 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)),Aa)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_1868_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,Aa: 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)),Aa)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_1869_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,Aa: extended_enat] :
      ( ord_le72135733267957522d_enat(B,Aa)
    <=> ( ( Aa = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B) )
        & ( Aa != B ) ) ) ).

% max.strict_order_iff
tff(fact_1870_max_Ostrict__order__iff,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),Aa)
    <=> ( ( Aa = ord_max_Code_integer(Aa,B) )
        & ( Aa != B ) ) ) ).

% max.strict_order_iff
tff(fact_1871_max_Ostrict__order__iff,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
    <=> ( ( Aa = aa_real_real(ord_max_real(Aa),B) )
        & ( Aa != B ) ) ) ).

% max.strict_order_iff
tff(fact_1872_max_Ostrict__order__iff,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
    <=> ( ( Aa = aa_rat_rat(ord_max_rat(Aa),B) )
        & ( Aa != B ) ) ) ).

% max.strict_order_iff
tff(fact_1873_max_Ostrict__order__iff,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
    <=> ( ( Aa = aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B) )
        & ( Aa != B ) ) ) ).

% max.strict_order_iff
tff(fact_1874_max_Ostrict__order__iff,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
    <=> ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) )
        & ( Aa != B ) ) ) ).

% max.strict_order_iff
tff(fact_1875_max_Ostrict__order__iff,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
    <=> ( ( Aa = aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B) )
        & ( Aa != B ) ) ) ).

% max.strict_order_iff
tff(fact_1876_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,Aa: extended_enat,B: extended_enat] :
      ( ord_le72135733267957522d_enat(C,Aa)
     => ord_le72135733267957522d_enat(C,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)) ) ).

% max.strict_coboundedI1
tff(fact_1877_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),Aa)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),ord_max_Code_integer(Aa,B)) ) ).

% max.strict_coboundedI1
tff(fact_1878_max_Ostrict__coboundedI1,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(ord_max_real(Aa),B)) ) ).

% max.strict_coboundedI1
tff(fact_1879_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(Aa),B)) ) ).

% max.strict_coboundedI1
tff(fact_1880_max_Ostrict__coboundedI1,axiom,
    ! [C: num,Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B)) ) ).

% max.strict_coboundedI1
tff(fact_1881_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B)) ) ).

% max.strict_coboundedI1
tff(fact_1882_max_Ostrict__coboundedI1,axiom,
    ! [C: int,Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,Aa),B)) ) ).

% max.strict_coboundedI1
tff(fact_1883_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,Aa: extended_enat] :
      ( ord_le72135733267957522d_enat(C,B)
     => ord_le72135733267957522d_enat(C,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Aa),B)) ) ).

% max.strict_coboundedI2
tff(fact_1884_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),ord_max_Code_integer(Aa,B)) ) ).

% max.strict_coboundedI2
tff(fact_1885_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,Aa: 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(ord_max_real(Aa),B)) ) ).

% max.strict_coboundedI2
tff(fact_1886_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(Aa),B)) ) ).

% max.strict_coboundedI2
tff(fact_1887_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),B)) ) ).

% max.strict_coboundedI2
tff(fact_1888_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,Aa: 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,Aa),B)) ) ).

% max.strict_coboundedI2
tff(fact_1889_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,Aa: 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,Aa),B)) ) ).

% max.strict_coboundedI2
tff(fact_1890_Diff__eq__empty__iff,axiom,
    ! [A: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A,B2) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A),B2) ) ).

% Diff_eq_empty_iff
tff(fact_1891_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat(A,B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A),B2) ) ).

% Diff_eq_empty_iff
tff(fact_1892_Diff__eq__empty__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A,B2) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A),B2) ) ).

% Diff_eq_empty_iff
tff(fact_1893_empty__subsetI,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A) ).

% empty_subsetI
tff(fact_1894_empty__subsetI,axiom,
    ! [A: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A) ).

% empty_subsetI
tff(fact_1895_empty__subsetI,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% empty_subsetI
tff(fact_1896_subset__empty,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1897_subset__empty,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_1898_subset__empty,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1899_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_1900_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_1901_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_1902_add__diff__cancel__right_H,axiom,
    ! [Aa: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Aa),B)),B) = Aa ).

% add_diff_cancel_right'
tff(fact_1903_add__diff__cancel__right_H,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),B)),B) = Aa ).

% add_diff_cancel_right'
tff(fact_1904_add__diff__cancel__right_H,axiom,
    ! [Aa: 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,Aa),B)),B) = Aa ).

% add_diff_cancel_right'
tff(fact_1905_add__diff__cancel__right_H,axiom,
    ! [Aa: 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,Aa),B)),B) = Aa ).

% add_diff_cancel_right'
tff(fact_1906_add__diff__cancel__right,axiom,
    ! [Aa: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(Aa),B) ).

% add_diff_cancel_right
tff(fact_1907_add__diff__cancel__right,axiom,
    ! [Aa: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(Aa),B) ).

% add_diff_cancel_right
tff(fact_1908_add__diff__cancel__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ).

% add_diff_cancel_right
tff(fact_1909_add__diff__cancel__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ).

% add_diff_cancel_right
tff(fact_1910_add__diff__cancel__left_H,axiom,
    ! [Aa: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Aa),B)),Aa) = B ).

% add_diff_cancel_left'
tff(fact_1911_add__diff__cancel__left_H,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),B)),Aa) = B ).

% add_diff_cancel_left'
tff(fact_1912_add__diff__cancel__left_H,axiom,
    ! [Aa: 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,Aa),B)),Aa) = B ).

% add_diff_cancel_left'
tff(fact_1913_add__diff__cancel__left_H,axiom,
    ! [Aa: 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,Aa),B)),Aa) = B ).

% add_diff_cancel_left'
tff(fact_1914_add__diff__cancel__left,axiom,
    ! [C: real,Aa: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(C),Aa)),aa_real_real(plus_plus_real(C),B)) = aa_real_real(minus_minus_real(Aa),B) ).

% add_diff_cancel_left
tff(fact_1915_add__diff__cancel__left,axiom,
    ! [C: rat,Aa: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),B)) = aa_rat_rat(minus_minus_rat(Aa),B) ).

% add_diff_cancel_left
tff(fact_1916_add__diff__cancel__left,axiom,
    ! [C: nat,Aa: 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),B) ).

% add_diff_cancel_left
tff(fact_1917_add__diff__cancel__left,axiom,
    ! [C: int,Aa: 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),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B) ).

% add_diff_cancel_left
tff(fact_1918_add__left__cancel,axiom,
    ! [Aa: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(Aa),B) = aa_real_real(plus_plus_real(Aa),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1919_add__left__cancel,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),B) = aa_rat_rat(plus_plus_rat(Aa),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1920_add__left__cancel,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1921_add__left__cancel,axiom,
    ! [Aa: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1922_add__right__cancel,axiom,
    ! [B: real,Aa: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),Aa) = aa_real_real(plus_plus_real(C),Aa) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1923_add__right__cancel,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),Aa) = aa_rat_rat(plus_plus_rat(C),Aa) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1924_add__right__cancel,axiom,
    ! [B: nat,Aa: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1925_add__right__cancel,axiom,
    ! [B: int,Aa: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1926_psubsetI,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( ( A != B2 )
       => aa_set_int_o(ord_less_set_int(A),B2) ) ) ).

% psubsetI
tff(fact_1927_subsetI,axiom,
    ! [A: set_complex,B2: set_complex] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A)
         => aa_set_complex_o(member_complex(X2),B2) )
     => aa_set_complex_o(ord_le211207098394363844omplex(A),B2) ) ).

% subsetI
tff(fact_1928_subsetI,axiom,
    ! [A: set_real,B2: set_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A)
         => aa_set_real_o(member_real(X2),B2) )
     => aa_set_real_o(ord_less_eq_set_real(A),B2) ) ).

% subsetI
tff(fact_1929_subsetI,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A)
         => aa_set_set_nat_o(member_set_nat(X2),B2) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2) ) ).

% subsetI
tff(fact_1930_subsetI,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set_nat_o(member_nat(X2),B2) )
     => aa_set_nat_o(ord_less_eq_set_nat(A),B2) ) ).

% subsetI
tff(fact_1931_subsetI,axiom,
    ! [A: set_int,B2: set_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => aa_set_int_o(member_int(X2),B2) )
     => aa_set_int_o(ord_less_eq_set_int(A),B2) ) ).

% subsetI
tff(fact_1932_subset__antisym,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A)
       => ( A = B2 ) ) ) ).

% subset_antisym
tff(fact_1933_add__le__cancel__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),Aa)),aa_real_real(plus_plus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ).

% add_le_cancel_left
tff(fact_1934_add__le__cancel__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ).

% add_le_cancel_left
tff(fact_1935_add__le__cancel__left,axiom,
    ! [C: nat,Aa: 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),Aa)),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,Aa),B) ) ).

% add_le_cancel_left
tff(fact_1936_add__le__cancel__left,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),B) ) ).

% add_le_cancel_left
tff(fact_1937_add__le__cancel__right,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ).

% add_le_cancel_right
tff(fact_1938_add__le__cancel__right,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ).

% add_le_cancel_right
tff(fact_1939_add__le__cancel__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ).

% add_le_cancel_right
tff(fact_1940_add__le__cancel__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ).

% add_le_cancel_right
tff(fact_1941_add__less__cancel__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),Aa)),aa_real_real(plus_plus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ).

% add_less_cancel_left
tff(fact_1942_add__less__cancel__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ).

% add_less_cancel_left
tff(fact_1943_add__less__cancel__left,axiom,
    ! [C: nat,Aa: 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B) ) ).

% add_less_cancel_left
tff(fact_1944_add__less__cancel__left,axiom,
    ! [C: int,Aa: 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),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B) ) ).

% add_less_cancel_left
tff(fact_1945_add__less__cancel__right,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ).

% add_less_cancel_right
tff(fact_1946_add__less__cancel__right,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ).

% add_less_cancel_right
tff(fact_1947_add__less__cancel__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ).

% add_less_cancel_right
tff(fact_1948_add__less__cancel__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ).

% add_less_cancel_right
tff(fact_1949_mult__1,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,one_one_complex),Aa) = Aa ).

% mult_1
tff(fact_1950_mult__1,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(one_one_real),Aa) = Aa ).

% mult_1
tff(fact_1951_mult__1,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(one_one_rat),Aa) = Aa ).

% mult_1
tff(fact_1952_mult__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Aa) = Aa ).

% mult_1
tff(fact_1953_mult__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),Aa) = Aa ).

% mult_1
tff(fact_1954_mult_Oright__neutral,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),one_one_complex) = Aa ).

% mult.right_neutral
tff(fact_1955_mult_Oright__neutral,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(Aa),one_one_real) = Aa ).

% mult.right_neutral
tff(fact_1956_mult_Oright__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(Aa),one_one_rat) = Aa ).

% mult.right_neutral
tff(fact_1957_mult_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),one_one_nat) = Aa ).

% mult.right_neutral
tff(fact_1958_mult_Oright__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),one_one_int) = Aa ).

% mult.right_neutral
tff(fact_1959_add__diff__cancel,axiom,
    ! [Aa: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Aa),B)),B) = Aa ).

% add_diff_cancel
tff(fact_1960_add__diff__cancel,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),B)),B) = Aa ).

% add_diff_cancel
tff(fact_1961_add__diff__cancel,axiom,
    ! [Aa: 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,Aa),B)),B) = Aa ).

% add_diff_cancel
tff(fact_1962_diff__add__cancel,axiom,
    ! [Aa: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(Aa),B)),B) = Aa ).

% diff_add_cancel
tff(fact_1963_diff__add__cancel,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(Aa),B)),B) = Aa ).

% diff_add_cancel
tff(fact_1964_diff__add__cancel,axiom,
    ! [Aa: 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,Aa),B)),B) = Aa ).

% diff_add_cancel
tff(fact_1965_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_1966_minus__set__def,axiom,
    ! [A: set_real,B2: set_real] : minus_minus_set_real(A,B2) = collect_real(minus_404528632214381484real_o(aTP_Lamm_a(A),aTP_Lamm_a(B2))) ).

% minus_set_def
tff(fact_1967_minus__set__def,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A,B2) = collec213857154873943460nt_int(minus_827889751674334651_int_o(aTP_Lamm_aa(A),aTP_Lamm_aa(B2))) ).

% minus_set_def
tff(fact_1968_minus__set__def,axiom,
    ! [A: set_complex,B2: set_complex] : minus_811609699411566653omplex(A,B2) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_ab(A),aTP_Lamm_ab(B2))) ).

% minus_set_def
tff(fact_1969_minus__set__def,axiom,
    ! [A: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A,B2) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ac(A),aTP_Lamm_ac(B2))) ).

% minus_set_def
tff(fact_1970_minus__set__def,axiom,
    ! [A: set_int,B2: set_int] : minus_minus_set_int(A,B2) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ae(A),aTP_Lamm_ae(B2))) ).

% minus_set_def
tff(fact_1971_minus__set__def,axiom,
    ! [A: set_nat,B2: set_nat] : minus_minus_set_nat(A,B2) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ad(A),aTP_Lamm_ad(B2))) ).

% minus_set_def
tff(fact_1972_set__diff__eq,axiom,
    ! [A: set_real,B2: set_real] : minus_minus_set_real(A,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_aq(A),B2)) ).

% set_diff_eq
tff(fact_1973_set__diff__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A,B2) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_ar(A),B2)) ).

% set_diff_eq
tff(fact_1974_set__diff__eq,axiom,
    ! [A: set_complex,B2: set_complex] : minus_811609699411566653omplex(A,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_as(A),B2)) ).

% set_diff_eq
tff(fact_1975_set__diff__eq,axiom,
    ! [A: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_at(A),B2)) ).

% set_diff_eq
tff(fact_1976_set__diff__eq,axiom,
    ! [A: set_int,B2: set_int] : minus_minus_set_int(A,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_au(A),B2)) ).

% set_diff_eq
tff(fact_1977_set__diff__eq,axiom,
    ! [A: set_nat,B2: set_nat] : minus_minus_set_nat(A,B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_av(A),B2)) ).

% set_diff_eq
tff(fact_1978_one__reorient,axiom,
    ! [Xb: complex] :
      ( ( one_one_complex = Xb )
    <=> ( Xb = one_one_complex ) ) ).

% one_reorient
tff(fact_1979_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_1980_one__reorient,axiom,
    ! [Xb: rat] :
      ( ( one_one_rat = Xb )
    <=> ( Xb = one_one_rat ) ) ).

% one_reorient
tff(fact_1981_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_1982_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_1983_mult_Oleft__commute,axiom,
    ! [B: real,Aa: real,C: real] : aa_real_real(times_times_real(B),aa_real_real(times_times_real(Aa),C)) = aa_real_real(times_times_real(Aa),aa_real_real(times_times_real(B),C)) ).

% mult.left_commute
tff(fact_1984_mult_Oleft__commute,axiom,
    ! [B: rat,Aa: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(Aa),C)) = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(times_times_rat(B),C)) ).

% mult.left_commute
tff(fact_1985_mult_Oleft__commute,axiom,
    ! [B: nat,Aa: 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,Aa),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.left_commute
tff(fact_1986_mult_Oleft__commute,axiom,
    ! [B: int,Aa: 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,Aa),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.left_commute
tff(fact_1987_mult_Ocommute,axiom,
    ! [Aa: real,B: real] : aa_real_real(times_times_real(Aa),B) = aa_real_real(times_times_real(B),Aa) ).

% mult.commute
tff(fact_1988_mult_Ocommute,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(times_times_rat(Aa),B) = aa_rat_rat(times_times_rat(B),Aa) ).

% mult.commute
tff(fact_1989_mult_Ocommute,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Aa) ).

% mult.commute
tff(fact_1990_mult_Ocommute,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),Aa) ).

% mult.commute
tff(fact_1991_mult_Oassoc,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(Aa),B)),C) = aa_real_real(times_times_real(Aa),aa_real_real(times_times_real(B),C)) ).

% mult.assoc
tff(fact_1992_mult_Oassoc,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Aa),B)),C) = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_1993_mult_Oassoc,axiom,
    ! [Aa: 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,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_1994_mult_Oassoc,axiom,
    ! [Aa: 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,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_1995_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(Aa),B)),C) = aa_real_real(times_times_real(Aa),aa_real_real(times_times_real(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_1996_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Aa),B)),C) = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_1997_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [Aa: 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,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_1998_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [Aa: 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,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_1999_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(Aa),B)),C) = aa_real_real(plus_plus_real(Aa),aa_real_real(plus_plus_real(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_2000_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Aa),B)),C) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_2001_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [Aa: 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),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_2002_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [Aa: 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),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_2003_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(plus_plus_real(I),K) = aa_real_real(plus_plus_real(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2004_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_2005_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_2006_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_2007_group__cancel_Oadd1,axiom,
    ! [A: real,K: real,Aa: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(K),Aa) )
     => ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(K),aa_real_real(plus_plus_real(Aa),B)) ) ) ).

% group_cancel.add1
tff(fact_2008_group__cancel_Oadd1,axiom,
    ! [A: rat,K: rat,Aa: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(K),Aa) )
     => ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(Aa),B)) ) ) ).

% group_cancel.add1
tff(fact_2009_group__cancel_Oadd1,axiom,
    ! [A: nat,K: nat,Aa: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Aa) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B)) ) ) ).

% group_cancel.add1
tff(fact_2010_group__cancel_Oadd1,axiom,
    ! [A: int,K: int,Aa: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),Aa) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B)) ) ) ).

% group_cancel.add1
tff(fact_2011_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,Aa: real] :
      ( ( B2 = aa_real_real(plus_plus_real(K),B) )
     => ( aa_real_real(plus_plus_real(Aa),B2) = aa_real_real(plus_plus_real(K),aa_real_real(plus_plus_real(Aa),B)) ) ) ).

% group_cancel.add2
tff(fact_2012_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,Aa: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(Aa),B2) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(Aa),B)) ) ) ).

% group_cancel.add2
tff(fact_2013_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,Aa: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B)) ) ) ).

% group_cancel.add2
tff(fact_2014_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,Aa: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B)) ) ) ).

% group_cancel.add2
tff(fact_2015_add_Oassoc,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(Aa),B)),C) = aa_real_real(plus_plus_real(Aa),aa_real_real(plus_plus_real(B),C)) ).

% add.assoc
tff(fact_2016_add_Oassoc,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Aa),B)),C) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.assoc
tff(fact_2017_add_Oassoc,axiom,
    ! [Aa: 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),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_2018_add_Oassoc,axiom,
    ! [Aa: 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),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_2019_add_Oleft__cancel,axiom,
    ! [Aa: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(Aa),B) = aa_real_real(plus_plus_real(Aa),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2020_add_Oleft__cancel,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),B) = aa_rat_rat(plus_plus_rat(Aa),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2021_add_Oleft__cancel,axiom,
    ! [Aa: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2022_add_Oright__cancel,axiom,
    ! [B: real,Aa: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),Aa) = aa_real_real(plus_plus_real(C),Aa) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2023_add_Oright__cancel,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),Aa) = aa_rat_rat(plus_plus_rat(C),Aa) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2024_add_Oright__cancel,axiom,
    ! [B: int,Aa: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2025_add_Ocommute,axiom,
    ! [Aa: real,B: real] : aa_real_real(plus_plus_real(Aa),B) = aa_real_real(plus_plus_real(B),Aa) ).

% add.commute
tff(fact_2026_add_Ocommute,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(plus_plus_rat(Aa),B) = aa_rat_rat(plus_plus_rat(B),Aa) ).

% add.commute
tff(fact_2027_add_Ocommute,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa) ).

% add.commute
tff(fact_2028_add_Ocommute,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa) ).

% add.commute
tff(fact_2029_add_Oleft__commute,axiom,
    ! [B: real,Aa: real,C: real] : aa_real_real(plus_plus_real(B),aa_real_real(plus_plus_real(Aa),C)) = aa_real_real(plus_plus_real(Aa),aa_real_real(plus_plus_real(B),C)) ).

% add.left_commute
tff(fact_2030_add_Oleft__commute,axiom,
    ! [B: rat,Aa: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(Aa),C)) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.left_commute
tff(fact_2031_add_Oleft__commute,axiom,
    ! [B: nat,Aa: 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,Aa),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.left_commute
tff(fact_2032_add_Oleft__commute,axiom,
    ! [B: int,Aa: 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,Aa),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.left_commute
tff(fact_2033_add__left__imp__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(Aa),B) = aa_real_real(plus_plus_real(Aa),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2034_add__left__imp__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),B) = aa_rat_rat(plus_plus_rat(Aa),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2035_add__left__imp__eq,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2036_add__left__imp__eq,axiom,
    ! [Aa: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2037_add__right__imp__eq,axiom,
    ! [B: real,Aa: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),Aa) = aa_real_real(plus_plus_real(C),Aa) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2038_add__right__imp__eq,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),Aa) = aa_rat_rat(plus_plus_rat(C),Aa) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2039_add__right__imp__eq,axiom,
    ! [B: nat,Aa: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2040_add__right__imp__eq,axiom,
    ! [B: int,Aa: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2041_subset__iff__psubset__eq,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
    <=> ( aa_set_int_o(ord_less_set_int(A),B2)
        | ( A = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_2042_subset__psubset__trans,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(ord_less_set_int(B2),C2)
       => aa_set_int_o(ord_less_set_int(A),C2) ) ) ).

% subset_psubset_trans
tff(fact_2043_subset__not__subset__eq,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B2)
        & ~ aa_set_int_o(ord_less_eq_set_int(B2),A) ) ) ).

% subset_not_subset_eq
tff(fact_2044_psubset__subset__trans,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),C2)
       => aa_set_int_o(ord_less_set_int(A),C2) ) ) ).

% psubset_subset_trans
tff(fact_2045_psubset__imp__subset,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
     => aa_set_int_o(ord_less_eq_set_int(A),B2) ) ).

% psubset_imp_subset
tff(fact_2046_psubset__eq,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B2)
        & ( A != B2 ) ) ) ).

% psubset_eq
tff(fact_2047_psubsetE,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A),B2)
         => aa_set_int_o(ord_less_eq_set_int(B2),A) ) ) ).

% psubsetE
tff(fact_2048_in__mono,axiom,
    ! [A: set_complex,B2: set_complex,Xb: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
     => ( aa_set_complex_o(member_complex(Xb),A)
       => aa_set_complex_o(member_complex(Xb),B2) ) ) ).

% in_mono
tff(fact_2049_in__mono,axiom,
    ! [A: set_real,B2: set_real,Xb: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B2)
     => ( aa_set_real_o(member_real(Xb),A)
       => aa_set_real_o(member_real(Xb),B2) ) ) ).

% in_mono
tff(fact_2050_in__mono,axiom,
    ! [A: set_set_nat,B2: set_set_nat,Xb: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2)
     => ( aa_set_set_nat_o(member_set_nat(Xb),A)
       => aa_set_set_nat_o(member_set_nat(Xb),B2) ) ) ).

% in_mono
tff(fact_2051_in__mono,axiom,
    ! [A: set_nat,B2: set_nat,Xb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
     => ( aa_set_nat_o(member_nat(Xb),A)
       => aa_set_nat_o(member_nat(Xb),B2) ) ) ).

% in_mono
tff(fact_2052_in__mono,axiom,
    ! [A: set_int,B2: set_int,Xb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(member_int(Xb),A)
       => aa_set_int_o(member_int(Xb),B2) ) ) ).

% in_mono
tff(fact_2053_subsetD,axiom,
    ! [A: set_complex,B2: set_complex,C: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
     => ( aa_set_complex_o(member_complex(C),A)
       => aa_set_complex_o(member_complex(C),B2) ) ) ).

% subsetD
tff(fact_2054_subsetD,axiom,
    ! [A: set_real,B2: set_real,C: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B2)
     => ( aa_set_real_o(member_real(C),A)
       => aa_set_real_o(member_real(C),B2) ) ) ).

% subsetD
tff(fact_2055_subsetD,axiom,
    ! [A: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2)
     => ( aa_set_set_nat_o(member_set_nat(C),A)
       => aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% subsetD
tff(fact_2056_subsetD,axiom,
    ! [A: set_nat,B2: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
     => ( aa_set_nat_o(member_nat(C),A)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% subsetD
tff(fact_2057_subsetD,axiom,
    ! [A: set_int,B2: set_int,C: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(member_int(C),A)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% subsetD
tff(fact_2058_equalityE,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = B2 )
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A),B2)
         => ~ aa_set_int_o(ord_less_eq_set_int(B2),A) ) ) ).

% equalityE
tff(fact_2059_subset__eq,axiom,
    ! [A: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
    <=> ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),A)
         => aa_set_complex_o(member_complex(X),B2) ) ) ).

% subset_eq
tff(fact_2060_subset__eq,axiom,
    ! [A: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B2)
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),A)
         => aa_set_real_o(member_real(X),B2) ) ) ).

% subset_eq
tff(fact_2061_subset__eq,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2)
    <=> ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),A)
         => aa_set_set_nat_o(member_set_nat(X),B2) ) ) ).

% subset_eq
tff(fact_2062_subset__eq,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A)
         => aa_set_nat_o(member_nat(X),B2) ) ) ).

% subset_eq
tff(fact_2063_subset__eq,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
    <=> ! [X: int] :
          ( aa_set_int_o(member_int(X),A)
         => aa_set_int_o(member_int(X),B2) ) ) ).

% subset_eq
tff(fact_2064_equalityD1,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = B2 )
     => aa_set_int_o(ord_less_eq_set_int(A),B2) ) ).

% equalityD1
tff(fact_2065_equalityD2,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = B2 )
     => aa_set_int_o(ord_less_eq_set_int(B2),A) ) ).

% equalityD2
tff(fact_2066_subset__iff,axiom,
    ! [A: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
    <=> ! [T: complex] :
          ( aa_set_complex_o(member_complex(T),A)
         => aa_set_complex_o(member_complex(T),B2) ) ) ).

% subset_iff
tff(fact_2067_subset__iff,axiom,
    ! [A: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B2)
    <=> ! [T: real] :
          ( aa_set_real_o(member_real(T),A)
         => aa_set_real_o(member_real(T),B2) ) ) ).

% subset_iff
tff(fact_2068_subset__iff,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2)
    <=> ! [T: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(T),A)
         => aa_set_set_nat_o(member_set_nat(T),B2) ) ) ).

% subset_iff
tff(fact_2069_subset__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat(T),A)
         => aa_set_nat_o(member_nat(T),B2) ) ) ).

% subset_iff
tff(fact_2070_subset__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
    <=> ! [T: int] :
          ( aa_set_int_o(member_int(T),A)
         => aa_set_int_o(member_int(T),B2) ) ) ).

% subset_iff
tff(fact_2071_subset__refl,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% subset_refl
tff(fact_2072_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_2073_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_2074_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X2: set_nat] :
          ( aa_set_nat_o(P,X2)
         => aa_set_nat_o(Q,X2) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_2075_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_2076_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_2077_subset__trans,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),C2)
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% subset_trans
tff(fact_2078_set__eq__subset,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = B2 )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B2)
        & aa_set_int_o(ord_less_eq_set_int(B2),A) ) ) ).

% set_eq_subset
tff(fact_2079_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_2080_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_2081_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q))
    <=> ! [X: set_nat] :
          ( aa_set_nat_o(P,X)
         => aa_set_nat_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_2082_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_2083_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_2084_Diff__mono,axiom,
    ! [A: set_nat,C2: set_nat,D2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D2),B2)
       => aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A,B2)),minus_minus_set_nat(C2,D2)) ) ) ).

% Diff_mono
tff(fact_2085_Diff__mono,axiom,
    ! [A: set_int,C2: set_int,D2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),C2)
     => ( aa_set_int_o(ord_less_eq_set_int(D2),B2)
       => aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A,B2)),minus_minus_set_int(C2,D2)) ) ) ).

% Diff_mono
tff(fact_2086_Diff__subset,axiom,
    ! [A: set_nat,B2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A,B2)),A) ).

% Diff_subset
tff(fact_2087_Diff__subset,axiom,
    ! [A: set_int,B2: set_int] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A,B2)),A) ).

% Diff_subset
tff(fact_2088_double__diff,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
       => ( minus_minus_set_nat(B2,minus_minus_set_nat(C2,A)) = A ) ) ) ).

% double_diff
tff(fact_2089_double__diff,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),C2)
       => ( minus_minus_set_int(B2,minus_minus_set_int(C2,A)) = A ) ) ) ).

% double_diff
tff(fact_2090_empty__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(aTP_Lamm_aw) ).

% empty_def
tff(fact_2091_empty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_ax) ).

% empty_def
tff(fact_2092_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_ay) ).

% empty_def
tff(fact_2093_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_az) ).

% empty_def
tff(fact_2094_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_ba) ).

% empty_def
tff(fact_2095_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_bb) ).

% empty_def
tff(fact_2096_less__eq__set__def,axiom,
    ! [A: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
    <=> ord_le7644322562001619171plex_o(aTP_Lamm_ab(A),aTP_Lamm_ab(B2)) ) ).

% less_eq_set_def
tff(fact_2097_less__eq__set__def,axiom,
    ! [A: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B2)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_a(A),aTP_Lamm_a(B2)) ) ).

% less_eq_set_def
tff(fact_2098_less__eq__set__def,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ac(A),aTP_Lamm_ac(B2)) ) ).

% less_eq_set_def
tff(fact_2099_less__eq__set__def,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ad(A),aTP_Lamm_ad(B2)) ) ).

% less_eq_set_def
tff(fact_2100_less__eq__set__def,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ae(A),aTP_Lamm_ae(B2)) ) ).

% less_eq_set_def
tff(fact_2101_Collect__subset,axiom,
    ! [A: set_real,P: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bc(A),P))),A) ).

% Collect_subset
tff(fact_2102_Collect__subset,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: fun_Pr3238374608701176162_int_o] : ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_bd(A),P)),A) ).

% Collect_subset
tff(fact_2103_Collect__subset,axiom,
    ! [A: set_complex,P: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A),P))),A) ).

% Collect_subset
tff(fact_2104_Collect__subset,axiom,
    ! [A: set_set_nat,P: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bf(A),P))),A) ).

% Collect_subset
tff(fact_2105_Collect__subset,axiom,
    ! [A: set_nat,P: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bg(A),P))),A) ).

% Collect_subset
tff(fact_2106_Collect__subset,axiom,
    ! [A: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(A),P))),A) ).

% Collect_subset
tff(fact_2107_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(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2108_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_2109_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_2110_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_2111_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(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2112_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_2113_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_2114_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_2115_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(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2116_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_2117_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_2118_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_2119_add__mono,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_mono
tff(fact_2120_add__mono,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_mono
tff(fact_2121_add__mono,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_2122_add__mono,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_2123_add__left__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),Aa)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_left_mono
tff(fact_2124_add__left__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_left_mono
tff(fact_2125_add__left__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_2126_add__left__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_2127_less__eqE,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C3) ) ).

% less_eqE
tff(fact_2128_add__right__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_right_mono
tff(fact_2129_add__right__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_right_mono
tff(fact_2130_add__right__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_2131_add__right__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_2132_le__iff__add,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
    <=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C4) ) ).

% le_iff_add
tff(fact_2133_add__le__imp__le__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),Aa)),aa_real_real(plus_plus_real(C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ).

% add_le_imp_le_left
tff(fact_2134_add__le__imp__le__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ).

% add_le_imp_le_left
tff(fact_2135_add__le__imp__le__left,axiom,
    ! [C: nat,Aa: 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),Aa)),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,Aa),B) ) ).

% add_le_imp_le_left
tff(fact_2136_add__le__imp__le__left,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),B) ) ).

% add_le_imp_le_left
tff(fact_2137_add__le__imp__le__right,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ).

% add_le_imp_le_right
tff(fact_2138_add__le__imp__le__right,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ).

% add_le_imp_le_right
tff(fact_2139_add__le__imp__le__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ).

% add_le_imp_le_right
tff(fact_2140_add__le__imp__le__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ).

% add_le_imp_le_right
tff(fact_2141_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(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2142_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_2143_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_2144_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_2145_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(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2146_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_2147_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_2148_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_2149_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(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2150_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_2151_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_2152_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_2153_add__strict__mono,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_strict_mono
tff(fact_2154_add__strict__mono,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(Aa),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_strict_mono
tff(fact_2155_add__strict__mono,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_2156_add__strict__mono,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_2157_add__strict__left__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),Aa)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_strict_left_mono
tff(fact_2158_add__strict__left__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_strict_left_mono
tff(fact_2159_add__strict__left__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_2160_add__strict__left__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_2161_add__strict__right__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_strict_right_mono
tff(fact_2162_add__strict__right__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_strict_right_mono
tff(fact_2163_add__strict__right__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_2164_add__strict__right__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_2165_add__less__imp__less__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),Aa)),aa_real_real(plus_plus_real(C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ).

% add_less_imp_less_left
tff(fact_2166_add__less__imp__less__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ).

% add_less_imp_less_left
tff(fact_2167_add__less__imp__less__left,axiom,
    ! [C: nat,Aa: 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B) ) ).

% add_less_imp_less_left
tff(fact_2168_add__less__imp__less__left,axiom,
    ! [C: int,Aa: 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),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B) ) ).

% add_less_imp_less_left
tff(fact_2169_add__less__imp__less__right,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ).

% add_less_imp_less_right
tff(fact_2170_add__less__imp__less__right,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ).

% add_less_imp_less_right
tff(fact_2171_add__less__imp__less__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ).

% add_less_imp_less_right
tff(fact_2172_add__less__imp__less__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ).

% add_less_imp_less_right
tff(fact_2173_diff__mono,axiom,
    ! [Aa: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(minus_minus_real(Aa),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_mono
tff(fact_2174_diff__mono,axiom,
    ! [Aa: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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,aa_rat_rat(minus_minus_rat(Aa),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_mono
tff(fact_2175_diff__mono,axiom,
    ! [Aa: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_2176_diff__left__mono,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(C),Aa)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_left_mono
tff(fact_2177_diff__left__mono,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),Aa)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_left_mono
tff(fact_2178_diff__left__mono,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_2179_diff__right__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Aa),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_right_mono
tff(fact_2180_diff__right__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Aa),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_right_mono
tff(fact_2181_diff__right__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_2182_diff__eq__diff__less__eq,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(Aa),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2183_diff__eq__diff__less__eq,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(Aa),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2184_diff__eq__diff__less__eq,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),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,Aa),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2185_diff__strict__right__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Aa),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_strict_right_mono
tff(fact_2186_diff__strict__right__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Aa),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_strict_right_mono
tff(fact_2187_diff__strict__right__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_2188_diff__strict__left__mono,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(C),Aa)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_strict_left_mono
tff(fact_2189_diff__strict__left__mono,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C),Aa)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_strict_left_mono
tff(fact_2190_diff__strict__left__mono,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_2191_diff__eq__diff__less,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(Aa),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_2192_diff__eq__diff__less,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(Aa),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_2193_diff__eq__diff__less,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),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,Aa),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_2194_diff__strict__mono,axiom,
    ! [Aa: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(minus_minus_real(Aa),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_strict_mono
tff(fact_2195_diff__strict__mono,axiom,
    ! [Aa: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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,aa_rat_rat(minus_minus_rat(Aa),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_strict_mono
tff(fact_2196_diff__strict__mono,axiom,
    ! [Aa: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_2197_mult_Ocomm__neutral,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),one_one_complex) = Aa ).

% mult.comm_neutral
tff(fact_2198_mult_Ocomm__neutral,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(Aa),one_one_real) = Aa ).

% mult.comm_neutral
tff(fact_2199_mult_Ocomm__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(Aa),one_one_rat) = Aa ).

% mult.comm_neutral
tff(fact_2200_mult_Ocomm__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),one_one_nat) = Aa ).

% mult.comm_neutral
tff(fact_2201_mult_Ocomm__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),one_one_int) = Aa ).

% mult.comm_neutral
tff(fact_2202_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,one_one_complex),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_2203_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(one_one_real),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_2204_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(one_one_rat),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_2205_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_2206_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_2207_group__cancel_Osub1,axiom,
    ! [A: real,K: real,Aa: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(K),Aa) )
     => ( aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(K),aa_real_real(minus_minus_real(Aa),B)) ) ) ).

% group_cancel.sub1
tff(fact_2208_group__cancel_Osub1,axiom,
    ! [A: rat,K: rat,Aa: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(K),Aa) )
     => ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(Aa),B)) ) ) ).

% group_cancel.sub1
tff(fact_2209_group__cancel_Osub1,axiom,
    ! [A: int,K: int,Aa: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),Aa) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B)) ) ) ).

% group_cancel.sub1
tff(fact_2210_diff__eq__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( ( aa_real_real(minus_minus_real(Aa),B) = C )
    <=> ( Aa = aa_real_real(plus_plus_real(C),B) ) ) ).

% diff_eq_eq
tff(fact_2211_diff__eq__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(Aa),B) = C )
    <=> ( Aa = aa_rat_rat(plus_plus_rat(C),B) ) ) ).

% diff_eq_eq
tff(fact_2212_diff__eq__eq,axiom,
    ! [Aa: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B) = C )
    <=> ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_2213_eq__diff__eq,axiom,
    ! [Aa: real,C: real,B: real] :
      ( ( Aa = aa_real_real(minus_minus_real(C),B) )
    <=> ( aa_real_real(plus_plus_real(Aa),B) = C ) ) ).

% eq_diff_eq
tff(fact_2214_eq__diff__eq,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( ( Aa = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(plus_plus_rat(Aa),B) = C ) ) ).

% eq_diff_eq
tff(fact_2215_eq__diff__eq,axiom,
    ! [Aa: int,C: int,B: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = C ) ) ).

% eq_diff_eq
tff(fact_2216_add__diff__eq,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(plus_plus_real(Aa),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Aa),B)),C) ).

% add_diff_eq
tff(fact_2217_add__diff__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),B)),C) ).

% add_diff_eq
tff(fact_2218_add__diff__eq,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),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,Aa),B)),C) ).

% add_diff_eq
tff(fact_2219_diff__diff__eq2,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(minus_minus_real(Aa),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Aa),C)),B) ).

% diff_diff_eq2
tff(fact_2220_diff__diff__eq2,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),C)),B) ).

% diff_diff_eq2
tff(fact_2221_diff__diff__eq2,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),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,Aa),C)),B) ).

% diff_diff_eq2
tff(fact_2222_diff__add__eq,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(Aa),B)),C) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Aa),C)),B) ).

% diff_add_eq
tff(fact_2223_diff__add__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(Aa),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),C)),B) ).

% diff_add_eq
tff(fact_2224_diff__add__eq,axiom,
    ! [Aa: 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,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),B) ).

% diff_add_eq
tff(fact_2225_diff__add__eq__diff__diff__swap,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(minus_minus_real(Aa),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(Aa),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_2226_diff__add__eq__diff__diff__swap,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(Aa),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_2227_diff__add__eq__diff__diff__swap,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),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,Aa),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_2228_add__implies__diff,axiom,
    ! [C: real,B: real,Aa: real] :
      ( ( aa_real_real(plus_plus_real(C),B) = Aa )
     => ( C = aa_real_real(minus_minus_real(Aa),B) ) ) ).

% add_implies_diff
tff(fact_2229_add__implies__diff,axiom,
    ! [C: rat,B: rat,Aa: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),B) = Aa )
     => ( C = aa_rat_rat(minus_minus_rat(Aa),B) ) ) ).

% add_implies_diff
tff(fact_2230_add__implies__diff,axiom,
    ! [C: nat,B: nat,Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = Aa )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),B) ) ) ).

% add_implies_diff
tff(fact_2231_add__implies__diff,axiom,
    ! [C: int,B: int,Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = Aa )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B) ) ) ).

% add_implies_diff
tff(fact_2232_diff__diff__eq,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(Aa),B)),C) = aa_real_real(minus_minus_real(Aa),aa_real_real(plus_plus_real(B),C)) ).

% diff_diff_eq
tff(fact_2233_diff__diff__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(Aa),B)),C) = aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(plus_plus_rat(B),C)) ).

% diff_diff_eq
tff(fact_2234_diff__diff__eq,axiom,
    ! [Aa: 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,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_2235_diff__diff__eq,axiom,
    ! [Aa: 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,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_2236_max__add__distrib__left,axiom,
    ! [Xb: real,Y3: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(ord_max_real(Xb),Y3)),Z) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(Xb),Z)),aa_real_real(plus_plus_real(Y3),Z)) ).

% max_add_distrib_left
tff(fact_2237_max__add__distrib__left,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(Xb),Y3)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Z)),aa_rat_rat(plus_plus_rat(Y3),Z)) ).

% max_add_distrib_left
tff(fact_2238_max__add__distrib__left,axiom,
    ! [Xb: nat,Y3: 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),Y3)),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,Y3),Z)) ).

% max_add_distrib_left
tff(fact_2239_max__add__distrib__left,axiom,
    ! [Xb: int,Y3: 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),Y3)),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,Y3),Z)) ).

% max_add_distrib_left
tff(fact_2240_max__add__distrib__left,axiom,
    ! [Xb: code_integer,Y3: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(ord_max_Code_integer(Xb,Y3),Z) = ord_max_Code_integer(plus_p5714425477246183910nteger(Xb,Z),plus_p5714425477246183910nteger(Y3,Z)) ).

% max_add_distrib_left
tff(fact_2241_max__add__distrib__right,axiom,
    ! [Xb: real,Y3: real,Z: real] : aa_real_real(plus_plus_real(Xb),aa_real_real(ord_max_real(Y3),Z)) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(Xb),Y3)),aa_real_real(plus_plus_real(Xb),Z)) ).

% max_add_distrib_right
tff(fact_2242_max__add__distrib__right,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] : aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(ord_max_rat(Y3),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Y3)),aa_rat_rat(plus_plus_rat(Xb),Z)) ).

% max_add_distrib_right
tff(fact_2243_max__add__distrib__right,axiom,
    ! [Xb: nat,Y3: 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,Y3),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)) ).

% max_add_distrib_right
tff(fact_2244_max__add__distrib__right,axiom,
    ! [Xb: int,Y3: 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,Y3),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)) ).

% max_add_distrib_right
tff(fact_2245_max__add__distrib__right,axiom,
    ! [Xb: code_integer,Y3: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(Xb,ord_max_Code_integer(Y3,Z)) = ord_max_Code_integer(plus_p5714425477246183910nteger(Xb,Y3),plus_p5714425477246183910nteger(Xb,Z)) ).

% max_add_distrib_right
tff(fact_2246_max__diff__distrib__left,axiom,
    ! [Xb: code_integer,Y3: code_integer,Z: code_integer] : minus_8373710615458151222nteger(ord_max_Code_integer(Xb,Y3),Z) = ord_max_Code_integer(minus_8373710615458151222nteger(Xb,Z),minus_8373710615458151222nteger(Y3,Z)) ).

% max_diff_distrib_left
tff(fact_2247_max__diff__distrib__left,axiom,
    ! [Xb: real,Y3: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(ord_max_real(Xb),Y3)),Z) = aa_real_real(ord_max_real(aa_real_real(minus_minus_real(Xb),Z)),aa_real_real(minus_minus_real(Y3),Z)) ).

% max_diff_distrib_left
tff(fact_2248_max__diff__distrib__left,axiom,
    ! [Xb: rat,Y3: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(Xb),Y3)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(Xb),Z)),aa_rat_rat(minus_minus_rat(Y3),Z)) ).

% max_diff_distrib_left
tff(fact_2249_max__diff__distrib__left,axiom,
    ! [Xb: int,Y3: 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),Y3)),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,Y3),Z)) ).

% max_diff_distrib_left
tff(fact_2250_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(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2251_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_2252_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_2253_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_2254_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(plus_plus_real(I),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2255_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_2256_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_2257_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_2258_add__le__less__mono,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_le_less_mono
tff(fact_2259_add__le__less__mono,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_le_less_mono
tff(fact_2260_add__le__less__mono,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2261_add__le__less__mono,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2262_add__less__le__mono,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(plus_plus_real(Aa),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_less_le_mono
tff(fact_2263_add__less__le__mono,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(Aa),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_less_le_mono
tff(fact_2264_add__less__le__mono,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2265_add__less__le__mono,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2266_diff__le__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Aa),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_le_eq
tff(fact_2267_diff__le__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Aa),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_le_eq
tff(fact_2268_diff__le__eq,axiom,
    ! [Aa: 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,Aa),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_2269_le__diff__eq,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Aa),B)),C) ) ).

% le_diff_eq
tff(fact_2270_le__diff__eq,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),B)),C) ) ).

% le_diff_eq
tff(fact_2271_le__diff__eq,axiom,
    ! [Aa: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),B)),C) ) ).

% le_diff_eq
tff(fact_2272_diff__add,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),Aa)),Aa) = B ) ) ).

% diff_add
tff(fact_2273_le__add__diff,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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)),Aa)) ) ).

% le_add_diff
tff(fact_2274_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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),Aa))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_2275_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),Aa) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_2276_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),Aa)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_2277_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),Aa)),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)),Aa) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_2278_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),Aa)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_2279_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_2280_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),Aa)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_2281_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),Aa) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_2282_diff__less__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Aa),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_less_eq
tff(fact_2283_diff__less__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Aa),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_less_eq
tff(fact_2284_diff__less__eq,axiom,
    ! [Aa: 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,Aa),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_2285_less__diff__eq,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Aa),B)),C) ) ).

% less_diff_eq
tff(fact_2286_less__diff__eq,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),B)),C) ) ).

% less_diff_eq
tff(fact_2287_less__diff__eq,axiom,
    ! [Aa: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),B)),C) ) ).

% less_diff_eq
tff(fact_2288_discrete,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),B) ) ).

% discrete
tff(fact_2289_discrete,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),one_one_int)),B) ) ).

% discrete
tff(fact_2290_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_2291_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_2292_divmod__step__eq,axiom,
    ! [L: num,Q2: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R2,aa_num_Code_integer(numera6620942414471956472nteger,L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_2293_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)),Aa) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),Aa)),C) ).

% times_divide_eq_left
tff(fact_2294_times__divide__eq__left,axiom,
    ! [B: real,C: real,Aa: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(B),C)),Aa) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(B),Aa)),C) ).

% times_divide_eq_left
tff(fact_2295_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,Aa: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B),C)),Aa) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B),Aa)),C) ).

% times_divide_eq_left
tff(fact_2296_divide__divide__eq__left,axiom,
    ! [Aa: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) ).

% divide_divide_eq_left
tff(fact_2297_divide__divide__eq__left,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(Aa),B)),C) = aa_real_real(divide_divide_real(Aa),aa_real_real(times_times_real(B),C)) ).

% divide_divide_eq_left
tff(fact_2298_divide__divide__eq__left,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(Aa),B)),C) = aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_2299_divide__divide__eq__right,axiom,
    ! [Aa: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C)),B) ).

% divide_divide_eq_right
tff(fact_2300_divide__divide__eq__right,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(divide_divide_real(Aa),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Aa),C)),B) ).

% divide_divide_eq_right
tff(fact_2301_divide__divide__eq__right,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),C)),B) ).

% divide_divide_eq_right
tff(fact_2302_times__divide__eq__right,axiom,
    ! [Aa: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B)),C) ).

% times_divide_eq_right
tff(fact_2303_times__divide__eq__right,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(times_times_real(Aa),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Aa),B)),C) ).

% times_divide_eq_right
tff(fact_2304_times__divide__eq__right,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),B)),C) ).

% times_divide_eq_right
tff(fact_2305_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,V),TreeLista,Vc)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_2306_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(Xb),Xaa) = Y3 )
     => ( ( ? [A2: $o,B3: $o] : Xb = vEBT_Leaf((A2),(B3))
         => ( ( Xaa = zero_zero_nat )
           => ( Y3 != one_one_nat ) ) )
       => ( ( ? [A2: $o,B3: $o] : Xb = vEBT_Leaf((A2),(B3))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y3 != one_one_nat ) ) )
         => ( ( ? [A2: $o,B3: $o] : Xb = vEBT_Leaf((A2),(B3))
             => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
               => ( Y3 != one_one_nat ) ) )
           => ( ( ? [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList,Summary)
                 => ( Y3 != one_one_nat ) )
               => ( ( ? [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = 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)
                   => ( Y3 != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = 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) )
                       => ( Y3 != 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(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(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_2307_delete__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_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_2308_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S)),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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_2309_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_2310_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_2311_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_2312_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_2313_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_2314_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_2315_Leaf__0__not,axiom,
    ! [Aa: $o,B: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((Aa),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_2316_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A3: $o,B4: $o] : Ta = vEBT_Leaf((A3),(B4)) ) ).

% deg1Leaf
tff(fact_2317_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
     => ? [A2: $o,B3: $o] : Ta = vEBT_Leaf((A2),(B3)) ) ).

% deg_1_Leaf
tff(fact_2318_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A2: $o,B3: $o] : Ta = vEBT_Leaf((A2),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_2319_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(Ta),Xb)
        | aa_nat_o(vEBT_VEBT_membermima(Ta),Xb) ) ) ).

% both_member_options_def
tff(fact_2320_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_2321_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xb)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_2322_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y222)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y222) ) ) ) ).

% VEBT.inject(2)
tff(fact_2323_delete__correct_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_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_2324_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_2325_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_2326_mult__zero__left,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,zero_zero_complex),Aa) = zero_zero_complex ).

% mult_zero_left
tff(fact_2327_mult__zero__left,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(zero_zero_real),Aa) = zero_zero_real ).

% mult_zero_left
tff(fact_2328_mult__zero__left,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),Aa) = zero_zero_rat ).

% mult_zero_left
tff(fact_2329_mult__zero__left,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% mult_zero_left
tff(fact_2330_mult__zero__left,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),Aa) = zero_zero_int ).

% mult_zero_left
tff(fact_2331_mult__zero__right,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_2332_mult__zero__right,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(Aa),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_2333_mult__zero__right,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(Aa),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_2334_mult__zero__right,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_2335_mult__zero__right,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_2336_mult__eq__0__iff,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B) = zero_zero_complex )
    <=> ( ( Aa = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_2337_mult__eq__0__iff,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(times_times_real(Aa),B) = zero_zero_real )
    <=> ( ( Aa = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_2338_mult__eq__0__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),B) = zero_zero_rat )
    <=> ( ( Aa = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_2339_mult__eq__0__iff,axiom,
    ! [Aa: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B) = zero_zero_nat )
    <=> ( ( Aa = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_2340_mult__eq__0__iff,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B) = zero_zero_int )
    <=> ( ( Aa = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_2341_mult__cancel__left,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),Aa) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( Aa = B ) ) ) ).

% mult_cancel_left
tff(fact_2342_mult__cancel__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( aa_real_real(times_times_real(C),Aa) = aa_real_real(times_times_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( Aa = B ) ) ) ).

% mult_cancel_left
tff(fact_2343_mult__cancel__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),Aa) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = B ) ) ) ).

% mult_cancel_left
tff(fact_2344_mult__cancel__left,axiom,
    ! [C: nat,Aa: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( Aa = B ) ) ) ).

% mult_cancel_left
tff(fact_2345_mult__cancel__left,axiom,
    ! [C: int,Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( Aa = B ) ) ) ).

% mult_cancel_left
tff(fact_2346_mult__cancel__right,axiom,
    ! [Aa: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( Aa = B ) ) ) ).

% mult_cancel_right
tff(fact_2347_mult__cancel__right,axiom,
    ! [Aa: real,C: real,B: real] :
      ( ( aa_real_real(times_times_real(Aa),C) = aa_real_real(times_times_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( Aa = B ) ) ) ).

% mult_cancel_right
tff(fact_2348_mult__cancel__right,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = B ) ) ) ).

% mult_cancel_right
tff(fact_2349_mult__cancel__right,axiom,
    ! [Aa: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( Aa = B ) ) ) ).

% mult_cancel_right
tff(fact_2350_mult__cancel__right,axiom,
    ! [Aa: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( Aa = B ) ) ) ).

% mult_cancel_right
tff(fact_2351_add__0,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),Aa) = Aa ).

% add_0
tff(fact_2352_add__0,axiom,
    ! [Aa: real] : aa_real_real(plus_plus_real(zero_zero_real),Aa) = Aa ).

% add_0
tff(fact_2353_add__0,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),Aa) = Aa ).

% add_0
tff(fact_2354_add__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Aa) = Aa ).

% add_0
tff(fact_2355_add__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Aa) = Aa ).

% add_0
tff(fact_2356_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y3) )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2357_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y3) = zero_zero_nat )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2358_add__cancel__right__right,axiom,
    ! [Aa: complex,B: complex] :
      ( ( Aa = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_2359_add__cancel__right__right,axiom,
    ! [Aa: real,B: real] :
      ( ( Aa = aa_real_real(plus_plus_real(Aa),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_2360_add__cancel__right__right,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa = aa_rat_rat(plus_plus_rat(Aa),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_2361_add__cancel__right__right,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_2362_add__cancel__right__right,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_2363_add__cancel__right__left,axiom,
    ! [Aa: complex,B: complex] :
      ( ( Aa = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),Aa) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_2364_add__cancel__right__left,axiom,
    ! [Aa: real,B: real] :
      ( ( Aa = aa_real_real(plus_plus_real(B),Aa) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_2365_add__cancel__right__left,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa = aa_rat_rat(plus_plus_rat(B),Aa) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_2366_add__cancel__right__left,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_2367_add__cancel__right__left,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_2368_add__cancel__left__right,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B) = Aa )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_2369_add__cancel__left__right,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(plus_plus_real(Aa),B) = Aa )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_2370_add__cancel__left__right,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),B) = Aa )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_2371_add__cancel__left__right,axiom,
    ! [Aa: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B) = Aa )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_2372_add__cancel__left__right,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = Aa )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_2373_add__cancel__left__left,axiom,
    ! [B: complex,Aa: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),Aa) = Aa )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_2374_add__cancel__left__left,axiom,
    ! [B: real,Aa: real] :
      ( ( aa_real_real(plus_plus_real(B),Aa) = Aa )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_2375_add__cancel__left__left,axiom,
    ! [B: rat,Aa: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),Aa) = Aa )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_2376_add__cancel__left__left,axiom,
    ! [B: nat,Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa) = Aa )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_2377_add__cancel__left__left,axiom,
    ! [B: int,Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa) = Aa )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_2378_double__zero__sym,axiom,
    ! [Aa: real] :
      ( ( zero_zero_real = aa_real_real(plus_plus_real(Aa),Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_2379_double__zero__sym,axiom,
    ! [Aa: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(Aa),Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_2380_double__zero__sym,axiom,
    ! [Aa: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Aa) )
    <=> ( Aa = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_2381_add_Oright__neutral,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),zero_zero_complex) = Aa ).

% add.right_neutral
tff(fact_2382_add_Oright__neutral,axiom,
    ! [Aa: real] : aa_real_real(plus_plus_real(Aa),zero_zero_real) = Aa ).

% add.right_neutral
tff(fact_2383_add_Oright__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(Aa),zero_zero_rat) = Aa ).

% add.right_neutral
tff(fact_2384_add_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),zero_zero_nat) = Aa ).

% add.right_neutral
tff(fact_2385_add_Oright__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),zero_zero_int) = Aa ).

% add.right_neutral
tff(fact_2386_division__ring__divide__zero,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2387_division__ring__divide__zero,axiom,
    ! [Aa: real] : aa_real_real(divide_divide_real(Aa),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_2388_division__ring__divide__zero,axiom,
    ! [Aa: rat] : aa_rat_rat(divide_divide_rat(Aa),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_2389_divide__cancel__right,axiom,
    ! [Aa: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( Aa = B ) ) ) ).

% divide_cancel_right
tff(fact_2390_divide__cancel__right,axiom,
    ! [Aa: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(Aa),C) = aa_real_real(divide_divide_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( Aa = B ) ) ) ).

% divide_cancel_right
tff(fact_2391_divide__cancel__right,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Aa),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = B ) ) ) ).

% divide_cancel_right
tff(fact_2392_divide__cancel__left,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,C),Aa) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( Aa = B ) ) ) ).

% divide_cancel_left
tff(fact_2393_divide__cancel__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( aa_real_real(divide_divide_real(C),Aa) = aa_real_real(divide_divide_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( Aa = B ) ) ) ).

% divide_cancel_left
tff(fact_2394_divide__cancel__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),Aa) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = B ) ) ) ).

% divide_cancel_left
tff(fact_2395_divide__eq__0__iff,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) = zero_zero_complex )
    <=> ( ( Aa = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_2396_divide__eq__0__iff,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(divide_divide_real(Aa),B) = zero_zero_real )
    <=> ( ( Aa = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_2397_divide__eq__0__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Aa),B) = zero_zero_rat )
    <=> ( ( Aa = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_2398_div__by__0,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_2399_div__by__0,axiom,
    ! [Aa: real] : aa_real_real(divide_divide_real(Aa),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_2400_div__by__0,axiom,
    ! [Aa: rat] : aa_rat_rat(divide_divide_rat(Aa),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_2401_div__by__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_2402_div__by__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_2403_div__0,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,zero_zero_complex),Aa) = zero_zero_complex ).

% div_0
tff(fact_2404_div__0,axiom,
    ! [Aa: real] : aa_real_real(divide_divide_real(zero_zero_real),Aa) = zero_zero_real ).

% div_0
tff(fact_2405_div__0,axiom,
    ! [Aa: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),Aa) = zero_zero_rat ).

% div_0
tff(fact_2406_div__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% div_0
tff(fact_2407_div__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),Aa) = zero_zero_int ).

% div_0
tff(fact_2408_bits__div__by__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_2409_bits__div__by__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_2410_bits__div__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% bits_div_0
tff(fact_2411_bits__div__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),Aa) = zero_zero_int ).

% bits_div_0
tff(fact_2412_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_2413_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_2414_bot__nat__0_Onot__eq__extremum,axiom,
    ! [Aa: nat] :
      ( ( Aa != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_2415_Nat_Oadd__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),zero_zero_nat) = Mb ).

% Nat.add_0_right
tff(fact_2416_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_2417_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_2418_bot__nat__0_Oextremum,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa) ).

% bot_nat_0.extremum
tff(fact_2419_insert__subset,axiom,
    ! [Xb: vEBT_VEBT,A: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(Xb,A)),B2)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),B2)
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B2) ) ) ).

% insert_subset
tff(fact_2420_insert__subset,axiom,
    ! [Xb: complex,A: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(insert_complex(Xb,A)),B2)
    <=> ( aa_set_complex_o(member_complex(Xb),B2)
        & aa_set_complex_o(ord_le211207098394363844omplex(A),B2) ) ) ).

% insert_subset
tff(fact_2421_insert__subset,axiom,
    ! [Xb: real,A: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(insert_real(Xb,A)),B2)
    <=> ( aa_set_real_o(member_real(Xb),B2)
        & aa_set_real_o(ord_less_eq_set_real(A),B2) ) ) ).

% insert_subset
tff(fact_2422_insert__subset,axiom,
    ! [Xb: set_nat,A: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat(Xb,A)),B2)
    <=> ( aa_set_set_nat_o(member_set_nat(Xb),B2)
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2) ) ) ).

% insert_subset
tff(fact_2423_insert__subset,axiom,
    ! [Xb: nat,A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(insert_nat(Xb,A)),B2)
    <=> ( aa_set_nat_o(member_nat(Xb),B2)
        & aa_set_nat_o(ord_less_eq_set_nat(A),B2) ) ) ).

% insert_subset
tff(fact_2424_insert__subset,axiom,
    ! [Xb: int,A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(insert_int(Xb,A)),B2)
    <=> ( aa_set_int_o(member_int(Xb),B2)
        & aa_set_int_o(ord_less_eq_set_int(A),B2) ) ) ).

% insert_subset
tff(fact_2425_mult__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2426_mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2427_mult__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_2428_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2429_diff__self__eq__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Mb) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_2430_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_2431_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_2432_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_2433_max__nat_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),zero_zero_nat) = Aa ).

% max_nat.right_neutral
tff(fact_2434_max__nat_Oneutr__eq__iff,axiom,
    ! [Aa: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) )
    <=> ( ( Aa = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_2435_max__nat_Oleft__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Aa) = Aa ).

% max_nat.left_neutral
tff(fact_2436_max__nat_Oeq__neutr__iff,axiom,
    ! [Aa: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),B) = zero_zero_nat )
    <=> ( ( Aa = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_2437_singleton__conv2,axiom,
    ! [Aa: vEBT_VEBT] : collect_VEBT_VEBT(fequal_VEBT_VEBT(Aa)) = insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv2
tff(fact_2438_singleton__conv2,axiom,
    ! [Aa: product_prod_int_int] : collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(fequal6010348034681699726nt_int,Aa)) = insert5033312907999012233nt_int(Aa,bot_bo1796632182523588997nt_int) ).

% singleton_conv2
tff(fact_2439_singleton__conv2,axiom,
    ! [Aa: complex] : collect_complex(aa_com5808801942499186519plex_o(fequal_complex,Aa)) = insert_complex(Aa,bot_bot_set_complex) ).

% singleton_conv2
tff(fact_2440_singleton__conv2,axiom,
    ! [Aa: set_nat] : collect_set_nat(fequal_set_nat(Aa)) = insert_set_nat(Aa,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_2441_singleton__conv2,axiom,
    ! [Aa: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,Aa)) = insert_nat(Aa,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_2442_singleton__conv2,axiom,
    ! [Aa: int] : collect_int(aa_int_fun_int_o(fequal_int,Aa)) = insert_int(Aa,bot_bot_set_int) ).

% singleton_conv2
tff(fact_2443_singleton__conv2,axiom,
    ! [Aa: real] : collect_real(aa_real_fun_real_o(fequal_real,Aa)) = insert_real(Aa,bot_bot_set_real) ).

% singleton_conv2
tff(fact_2444_singleton__conv,axiom,
    ! [Aa: vEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_bi(Aa)) = insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv
tff(fact_2445_singleton__conv,axiom,
    ! [Aa: product_prod_int_int] : collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(aTP_Lamm_bj,Aa)) = insert5033312907999012233nt_int(Aa,bot_bo1796632182523588997nt_int) ).

% singleton_conv
tff(fact_2446_singleton__conv,axiom,
    ! [Aa: complex] : collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_bk,Aa)) = insert_complex(Aa,bot_bot_set_complex) ).

% singleton_conv
tff(fact_2447_singleton__conv,axiom,
    ! [Aa: set_nat] : collect_set_nat(aTP_Lamm_bl(Aa)) = insert_set_nat(Aa,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_2448_singleton__conv,axiom,
    ! [Aa: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bm,Aa)) = insert_nat(Aa,bot_bot_set_nat) ).

% singleton_conv
tff(fact_2449_singleton__conv,axiom,
    ! [Aa: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_bn,Aa)) = insert_int(Aa,bot_bot_set_int) ).

% singleton_conv
tff(fact_2450_singleton__conv,axiom,
    ! [Aa: real] : collect_real(aa_real_fun_real_o(aTP_Lamm_bo,Aa)) = insert_real(Aa,bot_bot_set_real) ).

% singleton_conv
tff(fact_2451_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(Aa),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2452_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(Aa),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2453_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [Aa: 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),Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2454_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Aa),Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2455_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2456_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [Aa: 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),Aa)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2457_le__add__same__cancel2,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(plus_plus_real(B),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_2458_le__add__same__cancel2,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(plus_plus_rat(B),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_2459_le__add__same__cancel2,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_2460_le__add__same__cancel2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_2461_le__add__same__cancel1,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(plus_plus_real(Aa),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_2462_le__add__same__cancel1,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(plus_plus_rat(Aa),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_2463_le__add__same__cancel1,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_2464_le__add__same__cancel1,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_2465_add__le__same__cancel2,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Aa),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_2466_add__le__same__cancel2,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_2467_add__le__same__cancel2,axiom,
    ! [Aa: 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,Aa),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_2468_add__le__same__cancel2,axiom,
    ! [Aa: 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,Aa),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_2469_add__le__same__cancel1,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(B),Aa)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_2470_add__le__same__cancel1,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),Aa)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_2471_add__le__same__cancel1,axiom,
    ! [B: nat,Aa: 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),Aa)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_2472_add__le__same__cancel1,axiom,
    ! [B: int,Aa: 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),Aa)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_2473_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(Aa),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2474_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(Aa),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2475_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [Aa: 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),Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2476_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Aa),Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2477_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2478_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Aa)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2479_less__add__same__cancel2,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(plus_plus_real(B),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_2480_less__add__same__cancel2,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(plus_plus_rat(B),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_2481_less__add__same__cancel2,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_2482_less__add__same__cancel2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_2483_less__add__same__cancel1,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(plus_plus_real(Aa),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_2484_less__add__same__cancel1,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(plus_plus_rat(Aa),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_2485_less__add__same__cancel1,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_2486_less__add__same__cancel1,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_2487_add__less__same__cancel2,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Aa),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_2488_add__less__same__cancel2,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_2489_add__less__same__cancel2,axiom,
    ! [Aa: 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,Aa),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_2490_add__less__same__cancel2,axiom,
    ! [Aa: 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,Aa),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_2491_add__less__same__cancel1,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(B),Aa)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_2492_add__less__same__cancel1,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(B),Aa)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_2493_add__less__same__cancel1,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_2494_add__less__same__cancel1,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_2495_diff__ge__0__iff__ge,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(Aa),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa) ) ).

% diff_ge_0_iff_ge
tff(fact_2496_diff__ge__0__iff__ge,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(Aa),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ).

% diff_ge_0_iff_ge
tff(fact_2497_diff__ge__0__iff__ge,axiom,
    ! [Aa: 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,Aa),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa) ) ).

% diff_ge_0_iff_ge
tff(fact_2498_diff__gt__0__iff__gt,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(Aa),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa) ) ).

% diff_gt_0_iff_gt
tff(fact_2499_diff__gt__0__iff__gt,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(Aa),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa) ) ).

% diff_gt_0_iff_gt
tff(fact_2500_diff__gt__0__iff__gt,axiom,
    ! [Aa: 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,Aa),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa) ) ).

% diff_gt_0_iff_gt
tff(fact_2501_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_2502_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(times_times_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2503_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_2504_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_2505_mult__cancel__left2,axiom,
    ! [C: complex,Aa: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),Aa) = C )
    <=> ( ( C = zero_zero_complex )
        | ( Aa = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_2506_mult__cancel__left2,axiom,
    ! [C: real,Aa: real] :
      ( ( aa_real_real(times_times_real(C),Aa) = C )
    <=> ( ( C = zero_zero_real )
        | ( Aa = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2507_mult__cancel__left2,axiom,
    ! [C: rat,Aa: rat] :
      ( ( aa_rat_rat(times_times_rat(C),Aa) = C )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_2508_mult__cancel__left2,axiom,
    ! [C: int,Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) = C )
    <=> ( ( C = zero_zero_int )
        | ( Aa = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_2509_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_2510_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(times_times_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2511_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_2512_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_2513_mult__cancel__right2,axiom,
    ! [Aa: complex,C: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( Aa = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2514_mult__cancel__right2,axiom,
    ! [Aa: real,C: real] :
      ( ( aa_real_real(times_times_real(Aa),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( Aa = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2515_mult__cancel__right2,axiom,
    ! [Aa: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2516_mult__cancel__right2,axiom,
    ! [Aa: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( Aa = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_2517_sum__squares__eq__zero__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y3),Y3)) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2518_sum__squares__eq__zero__iff,axiom,
    ! [Xb: rat,Y3: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y3),Y3)) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2519_sum__squares__eq__zero__iff,axiom,
    ! [Xb: int,Y3: 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,Y3),Y3)) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2520_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_2521_diff__numeral__special_I9_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_2522_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_2523_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_2524_zero__eq__1__divide__iff,axiom,
    ! [Aa: real] :
      ( ( zero_zero_real = aa_real_real(divide_divide_real(one_one_real),Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2525_zero__eq__1__divide__iff,axiom,
    ! [Aa: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2526_one__divide__eq__0__iff,axiom,
    ! [Aa: real] :
      ( ( aa_real_real(divide_divide_real(one_one_real),Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2527_one__divide__eq__0__iff,axiom,
    ! [Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2528_eq__divide__eq__1,axiom,
    ! [B: real,Aa: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(B),Aa) )
    <=> ( ( Aa != zero_zero_real )
        & ( Aa = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2529_eq__divide__eq__1,axiom,
    ! [B: rat,Aa: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),Aa) )
    <=> ( ( Aa != zero_zero_rat )
        & ( Aa = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2530_divide__eq__eq__1,axiom,
    ! [B: real,Aa: real] :
      ( ( aa_real_real(divide_divide_real(B),Aa) = one_one_real )
    <=> ( ( Aa != zero_zero_real )
        & ( Aa = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2531_divide__eq__eq__1,axiom,
    ! [B: rat,Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),Aa) = one_one_rat )
    <=> ( ( Aa != zero_zero_rat )
        & ( Aa = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2532_divide__self__if,axiom,
    ! [Aa: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),Aa) = $ite(Aa = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_2533_divide__self__if,axiom,
    ! [Aa: real] :
      aa_real_real(divide_divide_real(Aa),Aa) = $ite(Aa = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_2534_divide__self__if,axiom,
    ! [Aa: rat] :
      aa_rat_rat(divide_divide_rat(Aa),Aa) = $ite(Aa = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_2535_divide__self,axiom,
    ! [Aa: complex] :
      ( ( Aa != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),Aa) = one_one_complex ) ) ).

% divide_self
tff(fact_2536_divide__self,axiom,
    ! [Aa: real] :
      ( ( Aa != zero_zero_real )
     => ( aa_real_real(divide_divide_real(Aa),Aa) = one_one_real ) ) ).

% divide_self
tff(fact_2537_divide__self,axiom,
    ! [Aa: rat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(Aa),Aa) = one_one_rat ) ) ).

% divide_self
tff(fact_2538_one__eq__divide__iff,axiom,
    ! [Aa: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) )
    <=> ( ( B != zero_zero_complex )
        & ( Aa = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2539_one__eq__divide__iff,axiom,
    ! [Aa: real,B: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(Aa),B) )
    <=> ( ( B != zero_zero_real )
        & ( Aa = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2540_one__eq__divide__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(Aa),B) )
    <=> ( ( B != zero_zero_rat )
        & ( Aa = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2541_divide__eq__1__iff,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( Aa = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2542_divide__eq__1__iff,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(divide_divide_real(Aa),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( Aa = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2543_divide__eq__1__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Aa),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( Aa = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2544_div__self,axiom,
    ! [Aa: complex] :
      ( ( Aa != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),Aa) = one_one_complex ) ) ).

% div_self
tff(fact_2545_div__self,axiom,
    ! [Aa: real] :
      ( ( Aa != zero_zero_real )
     => ( aa_real_real(divide_divide_real(Aa),Aa) = one_one_real ) ) ).

% div_self
tff(fact_2546_div__self,axiom,
    ! [Aa: rat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(Aa),Aa) = one_one_rat ) ) ).

% div_self
tff(fact_2547_div__self,axiom,
    ! [Aa: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Aa) = one_one_nat ) ) ).

% div_self
tff(fact_2548_div__self,axiom,
    ! [Aa: int] :
      ( ( Aa != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Aa) = one_one_int ) ) ).

% div_self
tff(fact_2549_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2550_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2551_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2552_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2553_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2554_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2555_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),Aa)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2556_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),Aa)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2557_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2558_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),Aa)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2559_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2560_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(Aa),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2561_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),Aa)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2562_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,Aa: real,B: real] :
      aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(divide_divide_real(Aa),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2563_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(Aa),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2564_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,Aa: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B)),B) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2565_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,Aa: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(Aa),B)),B) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2566_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,Aa: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),B)),B) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2567_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,Aa: 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,Aa),B)),B) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2568_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,Aa: 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,Aa),B)),B) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2569_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: complex,B: complex] :
      ( ( Aa != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B)),Aa) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2570_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: real,B: real] :
      ( ( Aa != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(Aa),B)),Aa) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2571_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),B)),Aa) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2572_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa != 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,Aa),B)),Aa) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2573_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa != 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),B)),Aa) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2574_div__mult__mult1,axiom,
    ! [C: nat,Aa: 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B) ) ) ).

% div_mult_mult1
tff(fact_2575_div__mult__mult1,axiom,
    ! [C: int,Aa: 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B) ) ) ).

% div_mult_mult1
tff(fact_2576_div__mult__mult2,axiom,
    ! [C: nat,Aa: 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,Aa),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,Aa),B) ) ) ).

% div_mult_mult2
tff(fact_2577_div__mult__mult2,axiom,
    ! [C: int,Aa: 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,Aa),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,Aa),B) ) ) ).

% div_mult_mult2
tff(fact_2578_div__mult__mult1__if,axiom,
    ! [C: nat,Aa: 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),Aa)),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,Aa),B)) ).

% div_mult_mult1_if
tff(fact_2579_div__mult__mult1__if,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),B)) ).

% div_mult_mult1_if
tff(fact_2580_diff__add__zero,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_2581_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_2582_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_2583_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_2584_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_2585_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2586_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_2587_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_2588_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_2589_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_2590_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_2591_power__Suc0__right,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(suc,zero_zero_nat)) = Aa ).

% power_Suc0_right
tff(fact_2592_power__Suc0__right,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(suc,zero_zero_nat)) = Aa ).

% power_Suc0_right
tff(fact_2593_power__Suc0__right,axiom,
    ! [Aa: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_nat_nat(suc,zero_zero_nat)) = Aa ).

% power_Suc0_right
tff(fact_2594_power__Suc0__right,axiom,
    ! [Aa: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(suc,zero_zero_nat)) = Aa ).

% power_Suc0_right
tff(fact_2595_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_2596_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_2597_add__gr__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_2598_div__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = Mb ).

% div_by_Suc_0
tff(fact_2599_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(3)
tff(fact_2600_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb)) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(3)
tff(fact_2601_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(zero_zero_real),aa_num_real(numeral_numeral_real,Xb)) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(3)
tff(fact_2602_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(3)
tff(fact_2603_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_2604_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_2605_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(Xb)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(4)
tff(fact_2606_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Xb),zero_z3403309356797280102nteger) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(4)
tff(fact_2607_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,Xb)),zero_zero_real) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(4)
tff(fact_2608_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xb)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(4)
tff(fact_2609_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_2610_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_2611_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_2612_max__0__1_I1_J,axiom,
    aa_real_real(ord_max_real(zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_2613_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_2614_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_2615_max__0__1_I1_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_2616_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_2617_max__0__1_I1_J,axiom,
    ord_max_Code_integer(zero_z3403309356797280102nteger,one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_2618_max__0__1_I2_J,axiom,
    aa_real_real(ord_max_real(one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_2619_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_2620_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_2621_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_2622_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_2623_max__0__1_I2_J,axiom,
    ord_max_Code_integer(one_one_Code_integer,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_2624_singleton__insert__inj__eq_H,axiom,
    ! [Aa: vEBT_VEBT,A: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( insert_VEBT_VEBT(Aa,A) = insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT) )
    <=> ( ( Aa = B )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2625_singleton__insert__inj__eq_H,axiom,
    ! [Aa: nat,A: set_nat,B: nat] :
      ( ( insert_nat(Aa,A) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( Aa = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2626_singleton__insert__inj__eq_H,axiom,
    ! [Aa: real,A: set_real,B: real] :
      ( ( insert_real(Aa,A) = insert_real(B,bot_bot_set_real) )
    <=> ( ( Aa = B )
        & aa_set_real_o(ord_less_eq_set_real(A),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2627_singleton__insert__inj__eq_H,axiom,
    ! [Aa: int,A: set_int,B: int] :
      ( ( insert_int(Aa,A) = insert_int(B,bot_bot_set_int) )
    <=> ( ( Aa = B )
        & aa_set_int_o(ord_less_eq_set_int(A),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2628_singleton__insert__inj__eq,axiom,
    ! [B: vEBT_VEBT,Aa: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT) = insert_VEBT_VEBT(Aa,A) )
    <=> ( ( Aa = B )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq
tff(fact_2629_singleton__insert__inj__eq,axiom,
    ! [B: nat,Aa: nat,A: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(Aa,A) )
    <=> ( ( Aa = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_2630_singleton__insert__inj__eq,axiom,
    ! [B: real,Aa: real,A: set_real] :
      ( ( insert_real(B,bot_bot_set_real) = insert_real(Aa,A) )
    <=> ( ( Aa = B )
        & aa_set_real_o(ord_less_eq_set_real(A),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq
tff(fact_2631_singleton__insert__inj__eq,axiom,
    ! [B: int,Aa: int,A: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(Aa,A) )
    <=> ( ( Aa = B )
        & aa_set_int_o(ord_less_eq_set_int(A),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_2632_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2633_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2634_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_2635_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2636_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_2637_mult__less__cancel2,axiom,
    ! [Mb: 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,Mb),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,Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_2638_zero__less__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% zero_less_diff
tff(fact_2639_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2640_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_2641_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_2642_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_2643_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_2644_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_2645_zero__le__divide__1__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% zero_le_divide_1_iff
tff(fact_2646_zero__le__divide__1__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% zero_le_divide_1_iff
tff(fact_2647_divide__le__0__1__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_2648_divide__le__0__1__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_2649_divide__less__0__1__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(one_one_real),Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_2650_divide__less__0__1__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_2651_divide__less__eq__1__neg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),Aa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2652_divide__less__eq__1__neg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),Aa)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2653_divide__less__eq__1__pos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),Aa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa) ) ) ).

% divide_less_eq_1_pos
tff(fact_2654_divide__less__eq__1__pos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),Aa)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa) ) ) ).

% divide_less_eq_1_pos
tff(fact_2655_less__divide__eq__1__neg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),Aa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa) ) ) ).

% less_divide_eq_1_neg
tff(fact_2656_less__divide__eq__1__neg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),Aa))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa) ) ) ).

% less_divide_eq_1_neg
tff(fact_2657_less__divide__eq__1__pos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),Aa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2658_less__divide__eq__1__pos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),Aa))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2659_zero__less__divide__1__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ).

% zero_less_divide_1_iff
tff(fact_2660_zero__less__divide__1__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ).

% zero_less_divide_1_iff
tff(fact_2661_eq__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: complex,B: complex,W: num] :
      ( ( Aa = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),numera6690914467698888265omplex(W)) = B,Aa = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2662_eq__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: real,B: real,W: num] :
      ( ( Aa = aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,W)) = B,Aa = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2663_eq__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: rat,B: rat,W: num] :
      ( ( Aa = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W)) = B,Aa = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2664_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,Aa: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),numera6690914467698888265omplex(W)) = Aa )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),numera6690914467698888265omplex(W)),Aa = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2665_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,Aa: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)) = Aa )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B = aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,W)),Aa = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2666_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) = Aa )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B = aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W)),Aa = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2667_nonzero__divide__mult__cancel__left,axiom,
    ! [Aa: complex,B: complex] :
      ( ( Aa != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2668_nonzero__divide__mult__cancel__left,axiom,
    ! [Aa: real,B: real] :
      ( ( Aa != zero_zero_real )
     => ( aa_real_real(divide_divide_real(Aa),aa_real_real(times_times_real(Aa),B)) = aa_real_real(divide_divide_real(one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2669_nonzero__divide__mult__cancel__left,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(times_times_rat(Aa),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2670_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,Aa: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),Aa) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2671_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,Aa: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(B),aa_real_real(times_times_real(Aa),B)) = aa_real_real(divide_divide_real(one_one_real),Aa) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2672_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,Aa: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(Aa),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),Aa) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2673_div__mult__self4,axiom,
    ! [B: nat,C: nat,Aa: 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)),Aa)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)) ) ) ).

% div_mult_self4
tff(fact_2674_div__mult__self4,axiom,
    ! [B: int,C: int,Aa: 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)),Aa)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)) ) ) ).

% div_mult_self4
tff(fact_2675_div__mult__self3,axiom,
    ! [B: nat,C: nat,Aa: 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)),Aa)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)) ) ) ).

% div_mult_self3
tff(fact_2676_div__mult__self3,axiom,
    ! [B: int,C: int,Aa: 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)),Aa)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)) ) ) ).

% div_mult_self3
tff(fact_2677_div__mult__self2,axiom,
    ! [B: nat,Aa: 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,Aa),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,Aa),B)) ) ) ).

% div_mult_self2
tff(fact_2678_div__mult__self2,axiom,
    ! [B: int,Aa: 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,Aa),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,Aa),B)) ) ) ).

% div_mult_self2
tff(fact_2679_div__mult__self1,axiom,
    ! [B: nat,Aa: 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,Aa),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,Aa),B)) ) ) ).

% div_mult_self1
tff(fact_2680_div__mult__self1,axiom,
    ! [B: int,Aa: 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,Aa),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,Aa),B)) ) ) ).

% div_mult_self1
tff(fact_2681_power__eq__0__iff,axiom,
    ! [Aa: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(Aa),Nb) = zero_zero_rat )
    <=> ( ( Aa = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2682_power__eq__0__iff,axiom,
    ! [Aa: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb) = zero_zero_nat )
    <=> ( ( Aa = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2683_power__eq__0__iff,axiom,
    ! [Aa: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb) = zero_zero_real )
    <=> ( ( Aa = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2684_power__eq__0__iff,axiom,
    ! [Aa: int,Nb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb) = zero_zero_int )
    <=> ( ( Aa = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2685_power__eq__0__iff,axiom,
    ! [Aa: complex,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb) = zero_zero_complex )
    <=> ( ( Aa = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2686_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_2687_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_2688_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2689_mult__le__cancel2,axiom,
    ! [Mb: 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,Mb),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,Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_2690_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_2691_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_2692_le__divide__eq__1__pos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),Aa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2693_le__divide__eq__1__pos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),Aa))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2694_le__divide__eq__1__neg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),Aa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa) ) ) ).

% le_divide_eq_1_neg
tff(fact_2695_le__divide__eq__1__neg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),Aa))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ) ).

% le_divide_eq_1_neg
tff(fact_2696_divide__le__eq__1__pos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),Aa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa) ) ) ).

% divide_le_eq_1_pos
tff(fact_2697_divide__le__eq__1__pos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),Aa)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ) ).

% divide_le_eq_1_pos
tff(fact_2698_divide__le__eq__1__neg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),Aa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2699_divide__le__eq__1__neg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),Aa)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2700_power__strict__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2701_power__strict__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2702_power__strict__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2703_power__strict__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Mb)),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),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2704_zero__eq__power2,axiom,
    ! [Aa: rat] :
      ( ( aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_2705_zero__eq__power2,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( Aa = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_2706_zero__eq__power2,axiom,
    ! [Aa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_2707_zero__eq__power2,axiom,
    ! [Aa: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_2708_zero__eq__power2,axiom,
    ! [Aa: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( Aa = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_2709_power__mono__iff,axiom,
    ! [Aa: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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,Aa),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,Aa),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2710_power__mono__iff,axiom,
    ! [Aa: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( 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(Aa),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2711_power__mono__iff,axiom,
    ! [Aa: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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,Aa),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,Aa),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2712_power__mono__iff,axiom,
    ! [Aa: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),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,Aa),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2713_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_2714_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_2715_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_2716_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_2717_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_2718_power2__eq__iff__nonneg,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( ( 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2719_power2__eq__iff__nonneg,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2720_power2__eq__iff__nonneg,axiom,
    ! [Xb: nat,Y3: 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),Y3)
       => ( ( 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2721_power2__eq__iff__nonneg,axiom,
    ! [Xb: int,Y3: 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),Y3)
       => ( ( 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2722_power2__less__eq__zero__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( Aa = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_2723_power2__less__eq__zero__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( Aa = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_2724_power2__less__eq__zero__iff,axiom,
    ! [Aa: 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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( Aa = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_2725_zero__less__power2,axiom,
    ! [Aa: 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( Aa != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_2726_zero__less__power2,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( Aa != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_2727_zero__less__power2,axiom,
    ! [Aa: 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( Aa != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_2728_power__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2729_power__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2730_power__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2731_power__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Mb)),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),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2732_sum__power2__eq__zero__iff,axiom,
    ! [Xb: rat,Y3: 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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2733_sum__power2__eq__zero__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2734_sum__power2__eq__zero__iff,axiom,
    ! [Xb: int,Y3: 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2735_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_2736_less__set__def,axiom,
    ! [A: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A),B2)
    <=> ord_le4409593196786536943plex_o(aTP_Lamm_ab(A),aTP_Lamm_ab(B2)) ) ).

% less_set_def
tff(fact_2737_less__set__def,axiom,
    ! [A: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A),B2)
    <=> ord_less_fun_real_o(aTP_Lamm_a(A),aTP_Lamm_a(B2)) ) ).

% less_set_def
tff(fact_2738_less__set__def,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A),B2)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ac(A),aTP_Lamm_ac(B2)) ) ).

% less_set_def
tff(fact_2739_less__set__def,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B2)
    <=> ord_less_fun_nat_o(aTP_Lamm_ad(A),aTP_Lamm_ad(B2)) ) ).

% less_set_def
tff(fact_2740_less__set__def,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
    <=> ord_less_fun_int_o(aTP_Lamm_ae(A),aTP_Lamm_ae(B2)) ) ).

% less_set_def
tff(fact_2741_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),D3)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_2742_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_2743_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv))),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_2744_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [Aa: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((Aa),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (Aa),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2745_vebt__delete_Osimps_I1_J,axiom,
    ! [Aa: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((Aa),(B)),zero_zero_nat) = vEBT_Leaf($false,(B)) ).

% vebt_delete.simps(1)
tff(fact_2746_insert__Collect,axiom,
    ! [Aa: vEBT_VEBT,P: fun_VEBT_VEBT_o] : insert_VEBT_VEBT(Aa,collect_VEBT_VEBT(P)) = collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_bp(Aa),P)) ).

% insert_Collect
tff(fact_2747_insert__Collect,axiom,
    ! [Aa: real,P: fun_real_o] : insert_real(Aa,collect_real(P)) = collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bq(Aa),P)) ).

% insert_Collect
tff(fact_2748_insert__Collect,axiom,
    ! [Aa: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] : insert5033312907999012233nt_int(Aa,collec213857154873943460nt_int(P)) = collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_br(Aa),P)) ).

% insert_Collect
tff(fact_2749_insert__Collect,axiom,
    ! [Aa: complex,P: fun_complex_o] : insert_complex(Aa,collect_complex(P)) = collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_bs(Aa),P)) ).

% insert_Collect
tff(fact_2750_insert__Collect,axiom,
    ! [Aa: set_nat,P: fun_set_nat_o] : insert_set_nat(Aa,collect_set_nat(P)) = collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bt(Aa),P)) ).

% insert_Collect
tff(fact_2751_insert__Collect,axiom,
    ! [Aa: nat,P: fun_nat_o] : insert_nat(Aa,collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bu(Aa),P)) ).

% insert_Collect
tff(fact_2752_insert__Collect,axiom,
    ! [Aa: int,P: fun_int_o] : insert_int(Aa,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bv(Aa),P)) ).

% insert_Collect
tff(fact_2753_insert__compr,axiom,
    ! [Aa: vEBT_VEBT,B2: set_VEBT_VEBT] : insert_VEBT_VEBT(Aa,B2) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_bw(Aa),B2)) ).

% insert_compr
tff(fact_2754_insert__compr,axiom,
    ! [Aa: real,B2: set_real] : insert_real(Aa,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_bx(Aa),B2)) ).

% insert_compr
tff(fact_2755_insert__compr,axiom,
    ! [Aa: product_prod_int_int,B2: set_Pr958786334691620121nt_int] : insert5033312907999012233nt_int(Aa,B2) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_by(Aa),B2)) ).

% insert_compr
tff(fact_2756_insert__compr,axiom,
    ! [Aa: complex,B2: set_complex] : insert_complex(Aa,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_bz(Aa),B2)) ).

% insert_compr
tff(fact_2757_insert__compr,axiom,
    ! [Aa: set_nat,B2: set_set_nat] : insert_set_nat(Aa,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_ca(Aa),B2)) ).

% insert_compr
tff(fact_2758_insert__compr,axiom,
    ! [Aa: nat,B2: set_nat] : insert_nat(Aa,B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_cb(Aa),B2)) ).

% insert_compr
tff(fact_2759_insert__compr,axiom,
    ! [Aa: int,B2: set_int] : insert_int(Aa,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_cc(Aa),B2)) ).

% insert_compr
tff(fact_2760_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A2: $o,B3: $o,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),X2)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_2761_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_2762_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_2763_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,A2: product_prod_nat_nat,B3: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A2),some_P7363390416028606310at_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_2764_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(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A2: nat,B3: nat] : Xb != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A2),some_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_2765_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(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A2: num,B3: num] : Xb != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A2),some_num(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_2766_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,Y: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X2),some_P7363390416028606310at_nat(Y))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_2767_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(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_o,X2: nat,Y: nat] : Xb != produc7967433646104525454on_nat(F2,produc5098337634421038937on_nat(some_nat(X2),some_nat(Y))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_2768_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(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_o,X2: num,Y: num] : Xb != produc1140783415258738358on_num(F2,produc8585076106096196333on_num(some_num(X2),some_num(Y))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_2769_invar__vebt_Ointros_I1_J,axiom,
    ! [Aa: $o,B: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((Aa),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_2770_vebt__delete_Osimps_I2_J,axiom,
    ! [Aa: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((Aa),(B)),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf((Aa),$false) ).

% vebt_delete.simps(2)
tff(fact_2771_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_2772_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_2773_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_2774_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_2775_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_2776_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_2777_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_2778_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_2779_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_2780_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2781_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,
    ! [Aa: $o,B: $o] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Leaf((Aa),(B))),zero_zero_nat) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_2782_vebt__member_Osimps_I1_J,axiom,
    ! [Aa: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((Aa),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (Aa),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_2783_VEBT_Oexhaust,axiom,
    ! [Y3: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y3 != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Y3 != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_2784_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_2785_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_2786_vebt__insert_Osimps_I1_J,axiom,
    ! [Aa: $o,B: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((Aa),(B)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(B)),
        $ite(Xb = one_one_nat,vEBT_Leaf((Aa),$true),vEBT_Leaf((Aa),(B))) ) ).

% vebt_insert.simps(1)
tff(fact_2787_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv)),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_2788_subset__insertI2,axiom,
    ! [A: set_nat,B2: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(B,B2)) ) ).

% subset_insertI2
tff(fact_2789_subset__insertI2,axiom,
    ! [A: set_VEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B2)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(B,B2)) ) ).

% subset_insertI2
tff(fact_2790_subset__insertI2,axiom,
    ! [A: set_real,B2: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B2)
     => aa_set_real_o(ord_less_eq_set_real(A),insert_real(B,B2)) ) ).

% subset_insertI2
tff(fact_2791_subset__insertI2,axiom,
    ! [A: set_int,B2: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => aa_set_int_o(ord_less_eq_set_int(A),insert_int(B,B2)) ) ).

% subset_insertI2
tff(fact_2792_subset__insertI,axiom,
    ! [B2: set_nat,Aa: nat] : aa_set_nat_o(ord_less_eq_set_nat(B2),insert_nat(Aa,B2)) ).

% subset_insertI
tff(fact_2793_subset__insertI,axiom,
    ! [B2: set_VEBT_VEBT,Aa: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B2),insert_VEBT_VEBT(Aa,B2)) ).

% subset_insertI
tff(fact_2794_subset__insertI,axiom,
    ! [B2: set_real,Aa: real] : aa_set_real_o(ord_less_eq_set_real(B2),insert_real(Aa,B2)) ).

% subset_insertI
tff(fact_2795_subset__insertI,axiom,
    ! [B2: set_int,Aa: int] : aa_set_int_o(ord_less_eq_set_int(B2),insert_int(Aa,B2)) ).

% subset_insertI
tff(fact_2796_subset__insert,axiom,
    ! [Xb: vEBT_VEBT,A: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(Xb,B2))
      <=> aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B2) ) ) ).

% subset_insert
tff(fact_2797_subset__insert,axiom,
    ! [Xb: complex,A: set_complex,B2: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xb),A)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),insert_complex(Xb,B2))
      <=> aa_set_complex_o(ord_le211207098394363844omplex(A),B2) ) ) ).

% subset_insert
tff(fact_2798_subset__insert,axiom,
    ! [Xb: real,A: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(Xb),A)
     => ( aa_set_real_o(ord_less_eq_set_real(A),insert_real(Xb,B2))
      <=> aa_set_real_o(ord_less_eq_set_real(A),B2) ) ) ).

% subset_insert
tff(fact_2799_subset__insert,axiom,
    ! [Xb: set_nat,A: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xb),A)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),insert_set_nat(Xb,B2))
      <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2) ) ) ).

% subset_insert
tff(fact_2800_subset__insert,axiom,
    ! [Xb: nat,A: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(Xb,B2))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A),B2) ) ) ).

% subset_insert
tff(fact_2801_subset__insert,axiom,
    ! [Xb: int,A: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(Xb),A)
     => ( aa_set_int_o(ord_less_eq_set_int(A),insert_int(Xb,B2))
      <=> aa_set_int_o(ord_less_eq_set_int(A),B2) ) ) ).

% subset_insert
tff(fact_2802_insert__mono,axiom,
    ! [C2: set_nat,D2: set_nat,Aa: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C2),D2)
     => aa_set_nat_o(ord_less_eq_set_nat(insert_nat(Aa,C2)),insert_nat(Aa,D2)) ) ).

% insert_mono
tff(fact_2803_insert__mono,axiom,
    ! [C2: set_VEBT_VEBT,D2: set_VEBT_VEBT,Aa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(C2),D2)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(Aa,C2)),insert_VEBT_VEBT(Aa,D2)) ) ).

% insert_mono
tff(fact_2804_insert__mono,axiom,
    ! [C2: set_real,D2: set_real,Aa: real] :
      ( aa_set_real_o(ord_less_eq_set_real(C2),D2)
     => aa_set_real_o(ord_less_eq_set_real(insert_real(Aa,C2)),insert_real(Aa,D2)) ) ).

% insert_mono
tff(fact_2805_insert__mono,axiom,
    ! [C2: set_int,D2: set_int,Aa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(C2),D2)
     => aa_set_int_o(ord_less_eq_set_int(insert_int(Aa,C2)),insert_int(Aa,D2)) ) ).

% insert_mono
tff(fact_2806_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(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_2807_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(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_2808_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(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_2809_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_2810_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_2811_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_2812_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_2813_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_2814_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_2815_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_2816_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_2817_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_2818_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_2819_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_2820_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_2821_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_2822_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_2823_gr__implies__not__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_2824_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_2825_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_2826_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).

% zero_neq_numeral
tff(fact_2827_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).

% zero_neq_numeral
tff(fact_2828_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_2829_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_2830_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2831_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2832_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2833_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2834_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2835_mult__not__zero,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B) != zero_zero_complex )
     => ( ( Aa != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_2836_mult__not__zero,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(times_times_real(Aa),B) != zero_zero_real )
     => ( ( Aa != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2837_mult__not__zero,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),B) != zero_zero_rat )
     => ( ( Aa != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2838_mult__not__zero,axiom,
    ! [Aa: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B) != zero_zero_nat )
     => ( ( Aa != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_2839_mult__not__zero,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B) != zero_zero_int )
     => ( ( Aa != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_2840_divisors__zero,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B) = zero_zero_complex )
     => ( ( Aa = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_2841_divisors__zero,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(times_times_real(Aa),B) = zero_zero_real )
     => ( ( Aa = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2842_divisors__zero,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),B) = zero_zero_rat )
     => ( ( Aa = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2843_divisors__zero,axiom,
    ! [Aa: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B) = zero_zero_nat )
     => ( ( Aa = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_2844_divisors__zero,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B) = zero_zero_int )
     => ( ( Aa = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_2845_no__zero__divisors,axiom,
    ! [Aa: complex,B: complex] :
      ( ( Aa != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_2846_no__zero__divisors,axiom,
    ! [Aa: real,B: real] :
      ( ( Aa != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(times_times_real(Aa),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2847_no__zero__divisors,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(Aa),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2848_no__zero__divisors,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_2849_no__zero__divisors,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_2850_mult__left__cancel,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),Aa) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) )
      <=> ( Aa = B ) ) ) ).

% mult_left_cancel
tff(fact_2851_mult__left__cancel,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(C),Aa) = aa_real_real(times_times_real(C),B) )
      <=> ( Aa = B ) ) ) ).

% mult_left_cancel
tff(fact_2852_mult__left__cancel,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),Aa) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( Aa = B ) ) ) ).

% mult_left_cancel
tff(fact_2853_mult__left__cancel,axiom,
    ! [C: nat,Aa: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( Aa = B ) ) ) ).

% mult_left_cancel
tff(fact_2854_mult__left__cancel,axiom,
    ! [C: int,Aa: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( Aa = B ) ) ) ).

% mult_left_cancel
tff(fact_2855_mult__right__cancel,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C) )
      <=> ( Aa = B ) ) ) ).

% mult_right_cancel
tff(fact_2856_mult__right__cancel,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(Aa),C) = aa_real_real(times_times_real(B),C) )
      <=> ( Aa = B ) ) ) ).

% mult_right_cancel
tff(fact_2857_mult__right__cancel,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(Aa),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( Aa = B ) ) ) ).

% mult_right_cancel
tff(fact_2858_mult__right__cancel,axiom,
    ! [C: nat,Aa: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( Aa = B ) ) ) ).

% mult_right_cancel
tff(fact_2859_mult__right__cancel,axiom,
    ! [C: int,Aa: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( Aa = B ) ) ) ).

% mult_right_cancel
tff(fact_2860_add_Ogroup__left__neutral,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),Aa) = Aa ).

% add.group_left_neutral
tff(fact_2861_add_Ogroup__left__neutral,axiom,
    ! [Aa: real] : aa_real_real(plus_plus_real(zero_zero_real),Aa) = Aa ).

% add.group_left_neutral
tff(fact_2862_add_Ogroup__left__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),Aa) = Aa ).

% add.group_left_neutral
tff(fact_2863_add_Ogroup__left__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Aa) = Aa ).

% add.group_left_neutral
tff(fact_2864_add_Ocomm__neutral,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),zero_zero_complex) = Aa ).

% add.comm_neutral
tff(fact_2865_add_Ocomm__neutral,axiom,
    ! [Aa: real] : aa_real_real(plus_plus_real(Aa),zero_zero_real) = Aa ).

% add.comm_neutral
tff(fact_2866_add_Ocomm__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(Aa),zero_zero_rat) = Aa ).

% add.comm_neutral
tff(fact_2867_add_Ocomm__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),zero_zero_nat) = Aa ).

% add.comm_neutral
tff(fact_2868_add_Ocomm__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),zero_zero_int) = Aa ).

% add.comm_neutral
tff(fact_2869_comm__monoid__add__class_Oadd__0,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),Aa) = Aa ).

% comm_monoid_add_class.add_0
tff(fact_2870_comm__monoid__add__class_Oadd__0,axiom,
    ! [Aa: real] : aa_real_real(plus_plus_real(zero_zero_real),Aa) = Aa ).

% comm_monoid_add_class.add_0
tff(fact_2871_comm__monoid__add__class_Oadd__0,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),Aa) = Aa ).

% comm_monoid_add_class.add_0
tff(fact_2872_comm__monoid__add__class_Oadd__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Aa) = Aa ).

% comm_monoid_add_class.add_0
tff(fact_2873_comm__monoid__add__class_Oadd__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Aa) = Aa ).

% comm_monoid_add_class.add_0
tff(fact_2874_power__not__zero,axiom,
    ! [Aa: rat,Nb: nat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(Aa),Nb) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_2875_power__not__zero,axiom,
    ! [Aa: nat,Nb: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2876_power__not__zero,axiom,
    ! [Aa: real,Nb: nat] :
      ( ( Aa != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2877_power__not__zero,axiom,
    ! [Aa: int,Nb: nat] :
      ( ( Aa != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2878_power__not__zero,axiom,
    ! [Aa: complex,Nb: nat] :
      ( ( Aa != zero_zero_complex )
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2879_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_2880_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_2881_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_2882_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_2883_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_2884_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% not0_implies_Suc
tff(fact_2885_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_not_Suc
tff(fact_2886_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_neq_Suc
tff(fact_2887_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_2888_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_2889_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X2),zero_zero_nat)
     => ( ! [Y: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y))
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_2890_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_2891_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y3 != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2892_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa_nat_nat(suc,X23) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_2893_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2894_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_2895_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero_nat != aa_nat_nat(suc,X23) ).

% nat.distinct(1)
tff(fact_2896_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
                  & ~ aa_nat_o(P,M) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_2897_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_2898_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_2899_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_2900_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_2901_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_2902_bot__nat__0_Oextremum__strict,axiom,
    ! [Aa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_2903_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = Mb )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_2904_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_2905_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_2906_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
     => ( Aa = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_2907_bot__nat__0_Oextremum__unique,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
    <=> ( Aa = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_2908_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_2909_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( K = zero_zero_nat )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2910_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_2911_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb) = zero_zero_nat )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_2912_minus__nat_Odiff__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),zero_zero_nat) = Mb ).

% minus_nat.diff_0
tff(fact_2913_divide__right__mono__neg,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(divide_divide_real(B),C)),aa_real_real(divide_divide_real(Aa),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2914_divide__right__mono__neg,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(Aa),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2915_divide__nonpos__nonpos,axiom,
    ! [Xb: real,Y3: 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,Y3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y3)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2916_divide__nonpos__nonpos,axiom,
    ! [Xb: rat,Y3: 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,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y3)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2917_divide__nonpos__nonneg,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y3)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_2918_divide__nonpos__nonneg,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_2919_divide__nonneg__nonpos,axiom,
    ! [Xb: real,Y3: 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,Y3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y3)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_2920_divide__nonneg__nonpos,axiom,
    ! [Xb: rat,Y3: 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,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_2921_divide__nonneg__nonneg,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y3)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2922_divide__nonneg__nonneg,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y3)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2923_zero__le__divide__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Aa),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & 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),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_2924_zero__le__divide__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & 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),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_2925_divide__right__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(divide_divide_real(Aa),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_right_mono
tff(fact_2926_divide__right__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(divide_divide_rat(Aa),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_right_mono
tff(fact_2927_divide__le__0__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Aa),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & 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),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_2928_divide__le__0__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Aa),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & 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),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_2929_divide__strict__right__mono__neg,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ( 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(divide_divide_real(Aa),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2930_divide__strict__right__mono__neg,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ( 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(divide_divide_rat(Aa),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2931_divide__strict__right__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(divide_divide_real(Aa),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2932_divide__strict__right__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(divide_divide_rat(Aa),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2933_zero__less__divide__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Aa),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & 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),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_2934_zero__less__divide__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & 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),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_2935_divide__less__cancel,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Aa),C)),aa_real_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,Aa),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),Aa) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2936_divide__less__cancel,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Aa),C)),aa_rat_rat(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,Aa),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),Aa) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2937_divide__less__0__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Aa),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & 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),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2938_divide__less__0__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Aa),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & 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),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2939_divide__pos__pos,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y3)) ) ) ).

% divide_pos_pos
tff(fact_2940_divide__pos__pos,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y3)) ) ) ).

% divide_pos_pos
tff(fact_2941_divide__pos__neg,axiom,
    ! [Xb: real,Y3: 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,Y3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Y3)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_2942_divide__pos__neg,axiom,
    ! [Xb: rat,Y3: 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,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_2943_divide__neg__pos,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Y3)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_2944_divide__neg__pos,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_2945_divide__neg__neg,axiom,
    ! [Xb: real,Y3: 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,Y3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y3)) ) ) ).

% divide_neg_neg
tff(fact_2946_divide__neg__neg,axiom,
    ! [Xb: rat,Y3: 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,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y3)) ) ) ).

% divide_neg_neg
tff(fact_2947_right__inverse__eq,axiom,
    ! [B: complex,Aa: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) = one_one_complex )
      <=> ( Aa = B ) ) ) ).

% right_inverse_eq
tff(fact_2948_right__inverse__eq,axiom,
    ! [B: real,Aa: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(Aa),B) = one_one_real )
      <=> ( Aa = B ) ) ) ).

% right_inverse_eq
tff(fact_2949_right__inverse__eq,axiom,
    ! [B: rat,Aa: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(Aa),B) = one_one_rat )
      <=> ( Aa = B ) ) ) ).

% right_inverse_eq
tff(fact_2950_nonzero__eq__divide__eq,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( Aa = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
      <=> ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2951_nonzero__eq__divide__eq,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( Aa = aa_real_real(divide_divide_real(B),C) )
      <=> ( aa_real_real(times_times_real(Aa),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2952_nonzero__eq__divide__eq,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( Aa = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(times_times_rat(Aa),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2953_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,Aa: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = Aa )
      <=> ( B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2954_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,Aa: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(B),C) = Aa )
      <=> ( B = aa_real_real(times_times_real(Aa),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2955_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,Aa: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = Aa )
      <=> ( B = aa_rat_rat(times_times_rat(Aa),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2956_eq__divide__imp,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C) = B )
       => ( Aa = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2957_eq__divide__imp,axiom,
    ! [C: real,Aa: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(Aa),C) = B )
       => ( Aa = aa_real_real(divide_divide_real(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2958_eq__divide__imp,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(Aa),C) = B )
       => ( Aa = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2959_divide__eq__imp,axiom,
    ! [C: complex,B: complex,Aa: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C) )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = Aa ) ) ) ).

% divide_eq_imp
tff(fact_2960_divide__eq__imp,axiom,
    ! [C: real,B: real,Aa: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(times_times_real(Aa),C) )
       => ( aa_real_real(divide_divide_real(B),C) = Aa ) ) ) ).

% divide_eq_imp
tff(fact_2961_divide__eq__imp,axiom,
    ! [C: rat,B: rat,Aa: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(Aa),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = Aa ) ) ) ).

% divide_eq_imp
tff(fact_2962_eq__divide__eq,axiom,
    ! [Aa: complex,B: complex,C: complex] :
      ( ( Aa = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C) = B,Aa = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_2963_eq__divide__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( ( Aa = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(Aa),C) = B,Aa = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_2964_eq__divide__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( ( Aa = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(Aa),C) = B,Aa = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_2965_divide__eq__eq,axiom,
    ! [B: complex,C: complex,Aa: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = Aa )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C),Aa = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_2966_divide__eq__eq,axiom,
    ! [B: real,C: real,Aa: real] :
      ( ( aa_real_real(divide_divide_real(B),C) = Aa )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(Aa),C),Aa = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_2967_divide__eq__eq,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = Aa )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(Aa),C),Aa = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_2968_frac__eq__eq,axiom,
    ! [Y3: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,W),Z) )
        <=> ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W),Y3) ) ) ) ) ).

% frac_eq_eq
tff(fact_2969_frac__eq__eq,axiom,
    ! [Y3: real,Z: real,Xb: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(divide_divide_real(Xb),Y3) = aa_real_real(divide_divide_real(W),Z) )
        <=> ( aa_real_real(times_times_real(Xb),Z) = aa_real_real(times_times_real(W),Y3) ) ) ) ) ).

% frac_eq_eq
tff(fact_2970_frac__eq__eq,axiom,
    ! [Y3: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(Xb),Y3) = aa_rat_rat(divide_divide_rat(W),Z) )
        <=> ( aa_rat_rat(times_times_rat(Xb),Z) = aa_rat_rat(times_times_rat(W),Y3) ) ) ) ) ).

% frac_eq_eq
tff(fact_2971_Collect__conv__if2,axiom,
    ! [Aa: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_cd(Aa),P)) = $ite(aa_VEBT_VEBT_o(P,Aa),insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if2
tff(fact_2972_Collect__conv__if2,axiom,
    ! [Aa: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_ce(Aa),P)) = $ite(aa_Pro4592068669816069595_int_o(P,Aa),insert5033312907999012233nt_int(Aa,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if2
tff(fact_2973_Collect__conv__if2,axiom,
    ! [Aa: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_cf(Aa),P)) = $ite(aa_complex_o(P,Aa),insert_complex(Aa,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if2
tff(fact_2974_Collect__conv__if2,axiom,
    ! [Aa: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_cg(Aa),P)) = $ite(aa_set_nat_o(P,Aa),insert_set_nat(Aa,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if2
tff(fact_2975_Collect__conv__if2,axiom,
    ! [Aa: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ch(Aa),P)) = $ite(aa_nat_o(P,Aa),insert_nat(Aa,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_2976_Collect__conv__if2,axiom,
    ! [Aa: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ci(Aa),P)) = $ite(aa_int_o(P,Aa),insert_int(Aa,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_2977_Collect__conv__if2,axiom,
    ! [Aa: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_cj(Aa),P)) = $ite(aa_real_o(P,Aa),insert_real(Aa,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if2
tff(fact_2978_Collect__conv__if,axiom,
    ! [Aa: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ck(Aa),P)) = $ite(aa_VEBT_VEBT_o(P,Aa),insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if
tff(fact_2979_Collect__conv__if,axiom,
    ! [Aa: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_cl(Aa),P)) = $ite(aa_Pro4592068669816069595_int_o(P,Aa),insert5033312907999012233nt_int(Aa,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if
tff(fact_2980_Collect__conv__if,axiom,
    ! [Aa: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_cm(Aa),P)) = $ite(aa_complex_o(P,Aa),insert_complex(Aa,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if
tff(fact_2981_Collect__conv__if,axiom,
    ! [Aa: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_cn(Aa),P)) = $ite(aa_set_nat_o(P,Aa),insert_set_nat(Aa,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if
tff(fact_2982_Collect__conv__if,axiom,
    ! [Aa: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_co(Aa),P)) = $ite(aa_nat_o(P,Aa),insert_nat(Aa,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_2983_Collect__conv__if,axiom,
    ! [Aa: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cp(Aa),P)) = $ite(aa_int_o(P,Aa),insert_int(Aa,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_2984_Collect__conv__if,axiom,
    ! [Aa: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_cq(Aa),P)) = $ite(aa_real_o(P,Aa),insert_real(Aa,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if
tff(fact_2985_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,Aa: 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),Aa)
       => ( 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,Aa),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
          <=> ( Aa = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2986_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,Aa: 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),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(Aa),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( Aa = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2987_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,Aa: 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),Aa)
       => ( 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,Aa),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( Aa = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2988_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,Aa: 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),Aa)
       => ( 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,Aa),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb) )
          <=> ( Aa = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2989_power__eq__imp__eq__base,axiom,
    ! [Aa: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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),Aa)
       => ( 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 = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2990_power__eq__imp__eq__base,axiom,
    ! [Aa: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(Aa),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
       => ( 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 = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2991_power__eq__imp__eq__base,axiom,
    ! [Aa: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),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),Aa)
       => ( 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 = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2992_power__eq__imp__eq__base,axiom,
    ! [Aa: int,Nb: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),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),Aa)
       => ( 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 = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2993_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,
    ! [Aa: $o,B: $o] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Leaf((Aa),(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_2994_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [Aa: $o,B: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Leaf((Aa),(B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((Aa),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_2995_lambda__zero,axiom,
    aTP_Lamm_cr = aa_com3342990543769683525omplex(times_times_complex,zero_zero_complex) ).

% lambda_zero
tff(fact_2996_lambda__zero,axiom,
    aTP_Lamm_cs = times_times_real(zero_zero_real) ).

% lambda_zero
tff(fact_2997_lambda__zero,axiom,
    aTP_Lamm_ct = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_2998_lambda__zero,axiom,
    aTP_Lamm_cu = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_2999_lambda__zero,axiom,
    aTP_Lamm_cv = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_3000_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [Aa: $o,Uw: $o] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf((Aa),(Uw))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_3001_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_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A2: $o,B3: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),zero_zero_nat)
     => ( ! [A2: $o,B3: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A2: $o,B3: $o,N: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))
         => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary),Uu2)
           => ( ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(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 != produc738532404422230701BT_nat(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 != produc738532404422230701BT_nat(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>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.cases
tff(fact_3002_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2),X2)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),X2)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3003_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] :
      ( ! [A2: $o,B3: $o,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),X2)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X2)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X2)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X2)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(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_3004_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] :
      ( ! [A2: $o,B3: $o,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),X2)
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S2),X2)
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2),X2)
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(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 != produc738532404422230701BT_nat(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_3005_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,B3: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B3)),zero_zero_nat)
     => ( ! [Uv2: $o,Uw2: $o,N: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va3)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Ve)
           => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(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_3006_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 != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)
     => ( ! [A2: $o,Uw2: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(Uw2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A2: $o,B3: $o,Va2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve),Vf)
             => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(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_3007_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,
    ! [Aa: $o,B: $o,Xb: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Leaf((Aa),(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_3008_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(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_3009_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),zero_zero_nat,Va,Vb)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3010_power__strict__mono,axiom,
    ! [Aa: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => ( 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,Aa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3011_power__strict__mono,axiom,
    ! [Aa: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
       => ( 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(Aa),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3012_power__strict__mono,axiom,
    ! [Aa: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
       => ( 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,Aa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3013_power__strict__mono,axiom,
    ! [Aa: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
       => ( 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,Aa),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3014_subset__singleton__iff,axiom,
    ! [X5: set_VEBT_VEBT,Aa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(X5),insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT))
    <=> ( ( X5 = bot_bo8194388402131092736T_VEBT )
        | ( X5 = insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singleton_iff
tff(fact_3015_subset__singleton__iff,axiom,
    ! [X5: set_nat,Aa: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X5),insert_nat(Aa,bot_bot_set_nat))
    <=> ( ( X5 = bot_bot_set_nat )
        | ( X5 = insert_nat(Aa,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_3016_subset__singleton__iff,axiom,
    ! [X5: set_real,Aa: real] :
      ( aa_set_real_o(ord_less_eq_set_real(X5),insert_real(Aa,bot_bot_set_real))
    <=> ( ( X5 = bot_bot_set_real )
        | ( X5 = insert_real(Aa,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_3017_subset__singleton__iff,axiom,
    ! [X5: set_int,Aa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X5),insert_int(Aa,bot_bot_set_int))
    <=> ( ( X5 = bot_bot_set_int )
        | ( X5 = insert_int(Aa,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_3018_subset__singletonD,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))
     => ( ( A = bot_bo8194388402131092736T_VEBT )
        | ( A = insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singletonD
tff(fact_3019_subset__singletonD,axiom,
    ! [A: set_nat,Xb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(Xb,bot_bot_set_nat))
     => ( ( A = bot_bot_set_nat )
        | ( A = insert_nat(Xb,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_3020_subset__singletonD,axiom,
    ! [A: set_real,Xb: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),insert_real(Xb,bot_bot_set_real))
     => ( ( A = bot_bot_set_real )
        | ( A = insert_real(Xb,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_3021_subset__singletonD,axiom,
    ! [A: set_int,Xb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),insert_int(Xb,bot_bot_set_int))
     => ( ( A = bot_bot_set_int )
        | ( A = insert_int(Xb,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_3022_vebt__delete_Osimps_I3_J,axiom,
    ! [Aa: $o,B: $o,Nb: nat] : vEBT_vebt_delete(vEBT_Leaf((Aa),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = vEBT_Leaf((Aa),(B)) ).

% vebt_delete.simps(3)
tff(fact_3023_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [Aa: $o,Uw: $o] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((Aa),(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_3024_vebt__mint_Osimps_I1_J,axiom,
    ! [Aa: $o,B: $o] :
      vEBT_vebt_mint(vEBT_Leaf((Aa),(B))) = $ite(
        (Aa),
        some_nat(zero_zero_nat),
        $ite((B),some_nat(one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_3025_field__le__epsilon,axiom,
    ! [Xb: real,Y3: 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(plus_plus_real(Y3),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ).

% field_le_epsilon
tff(fact_3026_field__le__epsilon,axiom,
    ! [Xb: rat,Y3: 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(Y3),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3) ) ).

% field_le_epsilon
tff(fact_3027_vebt__maxt_Osimps_I1_J,axiom,
    ! [Aa: $o,B: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((Aa),(B))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((Aa),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_3028_subset__Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,B2: set_VEBT_VEBT,Xb: vEBT_VEBT,C2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),minus_5127226145743854075T_VEBT(B2,insert_VEBT_VEBT(Xb,C2)))
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),minus_5127226145743854075T_VEBT(B2,C2))
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A) ) ) ).

% subset_Diff_insert
tff(fact_3029_subset__Diff__insert,axiom,
    ! [A: set_complex,B2: set_complex,Xb: complex,C2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),minus_811609699411566653omplex(B2,insert_complex(Xb,C2)))
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(A),minus_811609699411566653omplex(B2,C2))
        & ~ aa_set_complex_o(member_complex(Xb),A) ) ) ).

% subset_Diff_insert
tff(fact_3030_subset__Diff__insert,axiom,
    ! [A: set_real,B2: set_real,Xb: real,C2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),minus_minus_set_real(B2,insert_real(Xb,C2)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A),minus_minus_set_real(B2,C2))
        & ~ aa_set_real_o(member_real(Xb),A) ) ) ).

% subset_Diff_insert
tff(fact_3031_subset__Diff__insert,axiom,
    ! [A: set_set_nat,B2: set_set_nat,Xb: set_nat,C2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),minus_2163939370556025621et_nat(B2,insert_set_nat(Xb,C2)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),minus_2163939370556025621et_nat(B2,C2))
        & ~ aa_set_set_nat_o(member_set_nat(Xb),A) ) ) ).

% subset_Diff_insert
tff(fact_3032_subset__Diff__insert,axiom,
    ! [A: set_nat,B2: set_nat,Xb: nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),minus_minus_set_nat(B2,insert_nat(Xb,C2)))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),minus_minus_set_nat(B2,C2))
        & ~ aa_set_nat_o(member_nat(Xb),A) ) ) ).

% subset_Diff_insert
tff(fact_3033_subset__Diff__insert,axiom,
    ! [A: set_int,B2: set_int,Xb: int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),minus_minus_set_int(B2,insert_int(Xb,C2)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),minus_minus_set_int(B2,C2))
        & ~ aa_set_int_o(member_int(Xb),A) ) ) ).

% subset_Diff_insert
tff(fact_3034_divide__nonpos__pos,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y3)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_3035_divide__nonpos__pos,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3036_divide__nonpos__neg,axiom,
    ! [Xb: real,Y3: 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,Y3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y3)) ) ) ).

% divide_nonpos_neg
tff(fact_3037_divide__nonpos__neg,axiom,
    ! [Xb: rat,Y3: 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,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y3)) ) ) ).

% divide_nonpos_neg
tff(fact_3038_divide__nonneg__pos,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y3)) ) ) ).

% divide_nonneg_pos
tff(fact_3039_divide__nonneg__pos,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y3)) ) ) ).

% divide_nonneg_pos
tff(fact_3040_divide__nonneg__neg,axiom,
    ! [Xb: real,Y3: 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,Y3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y3)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_3041_divide__nonneg__neg,axiom,
    ! [Xb: rat,Y3: 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,Y3),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3042_divide__le__cancel,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Aa),C)),aa_real_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,Aa),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),Aa) ) ) ) ).

% divide_le_cancel
tff(fact_3043_divide__le__cancel,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Aa),C)),aa_rat_rat(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,Aa),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),Aa) ) ) ) ).

% divide_le_cancel
tff(fact_3044_frac__less2,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( 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(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Y3),W)) ) ) ) ) ).

% frac_less2
tff(fact_3045_frac__less2,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => ( 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,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y3),W)) ) ) ) ) ).

% frac_less2
tff(fact_3046_frac__less,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( 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(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Y3),W)) ) ) ) ) ).

% frac_less
tff(fact_3047_frac__less,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => ( 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,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y3),W)) ) ) ) ) ).

% frac_less
tff(fact_3048_frac__le,axiom,
    ! [Y3: real,Xb: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
       => ( 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(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Y3),W)) ) ) ) ) ).

% frac_le
tff(fact_3049_frac__le,axiom,
    ! [Y3: rat,Xb: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
       => ( 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,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y3),W)) ) ) ) ) ).

% frac_le
tff(fact_3050_divide__less__eq__1,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),Aa)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) )
        | ( Aa = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_3051_divide__less__eq__1,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),Aa)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) )
        | ( Aa = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3052_less__divide__eq__1,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),Aa))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa) ) ) ) ).

% less_divide_eq_1
tff(fact_3053_less__divide__eq__1,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),Aa))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa) ) ) ) ).

% less_divide_eq_1
tff(fact_3054_divide__strict__left__mono__neg,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(times_times_real(Aa),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),Aa)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3055_divide__strict__left__mono__neg,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(Aa),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),Aa)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3056_divide__strict__left__mono,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ( 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(times_times_real(Aa),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),Aa)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3057_divide__strict__left__mono,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ( 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(Aa),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),Aa)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3058_mult__imp__less__div__pos,axiom,
    ! [Y3: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Z),Y3)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(divide_divide_real(Xb),Y3)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3059_mult__imp__less__div__pos,axiom,
    ! [Y3: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y3)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Y3)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3060_mult__imp__div__pos__less,axiom,
    ! [Y3: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(times_times_real(Z),Y3))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Y3)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3061_mult__imp__div__pos__less,axiom,
    ! [Y3: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(times_times_rat(Z),Y3))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3062_pos__less__divide__eq,axiom,
    ! [C: real,Aa: 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),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3063_pos__less__divide__eq,axiom,
    ! [C: rat,Aa: 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),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3064_pos__divide__less__eq,axiom,
    ! [C: real,B: real,Aa: 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(divide_divide_real(B),C)),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(Aa),C)) ) ) ).

% pos_divide_less_eq
tff(fact_3065_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,Aa: 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(divide_divide_rat(B),C)),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% pos_divide_less_eq
tff(fact_3066_neg__less__divide__eq,axiom,
    ! [C: real,Aa: 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),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(Aa),C)) ) ) ).

% neg_less_divide_eq
tff(fact_3067_neg__less__divide__eq,axiom,
    ! [C: rat,Aa: 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),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% neg_less_divide_eq
tff(fact_3068_neg__divide__less__eq,axiom,
    ! [C: real,B: real,Aa: 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(divide_divide_real(B),C)),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3069_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,Aa: 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(divide_divide_rat(B),C)),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3070_less__divide__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_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(times_times_real(Aa),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(times_times_real(Aa),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_3071_less__divide__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),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(Aa),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_3072_divide__less__eq,axiom,
    ! [B: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),Aa)
    <=> $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(times_times_real(Aa),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(times_times_real(Aa),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)) ) ) ).

% divide_less_eq
tff(fact_3073_divide__less__eq,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),Aa)
    <=> $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(Aa),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(Aa),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)) ) ) ).

% divide_less_eq
tff(fact_3074_divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z)),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3075_divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),Z)),Y3) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(Y3),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3076_divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Y3) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Y3),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3077_add__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Y3),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),Y3)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3078_add__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(Y3),Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Z)),Y3)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3079_add__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(Y3),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Y3)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3080_add__num__frac,axiom,
    ! [Y3: complex,Z: complex,Xb: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),Y3))),Y3) ) ) ).

% add_num_frac
tff(fact_3081_add__num__frac,axiom,
    ! [Y3: real,Z: real,Xb: real] :
      ( ( Y3 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(Xb),Y3)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(Z),Y3))),Y3) ) ) ).

% add_num_frac
tff(fact_3082_add__num__frac,axiom,
    ! [Y3: rat,Z: rat,Xb: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(Xb),Y3)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y3))),Y3) ) ) ).

% add_num_frac
tff(fact_3083_add__frac__num,axiom,
    ! [Y3: complex,Xb: complex,Z: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3)),Z) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),Y3))),Y3) ) ) ).

% add_frac_num
tff(fact_3084_add__frac__num,axiom,
    ! [Y3: real,Xb: real,Z: real] :
      ( ( Y3 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),Y3)),Z) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(Z),Y3))),Y3) ) ) ).

% add_frac_num
tff(fact_3085_add__frac__num,axiom,
    ! [Y3: rat,Xb: rat,Z: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Y3)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y3))),Y3) ) ) ).

% add_frac_num
tff(fact_3086_add__frac__eq,axiom,
    ! [Y3: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,W),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W),Y3))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3087_add__frac__eq,axiom,
    ! [Y3: real,Z: real,Xb: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),Y3)),aa_real_real(divide_divide_real(W),Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Y3))),aa_real_real(times_times_real(Y3),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3088_add__frac__eq,axiom,
    ! [Y3: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Y3)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y3))),aa_rat_rat(times_times_rat(Y3),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3089_add__divide__eq__if__simps_I1_J,axiom,
    ! [Aa: complex,B: complex,Z: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),Z)) = $ite(Z = zero_zero_complex,Aa,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3090_add__divide__eq__if__simps_I1_J,axiom,
    ! [Aa: real,B: real,Z: real] :
      aa_real_real(plus_plus_real(Aa),aa_real_real(divide_divide_real(B),Z)) = $ite(Z = zero_zero_real,Aa,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3091_add__divide__eq__if__simps_I1_J,axiom,
    ! [Aa: rat,B: rat,Z: rat] :
      aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,Aa,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3092_add__divide__eq__if__simps_I2_J,axiom,
    ! [Aa: complex,Z: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),Z)),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3093_add__divide__eq__if__simps_I2_J,axiom,
    ! [Aa: real,Z: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Aa),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Aa),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3094_add__divide__eq__if__simps_I2_J,axiom,
    ! [Aa: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Aa),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3095_vebt__pred_Osimps_I2_J,axiom,
    ! [Aa: $o,Uw: $o] :
      vEBT_vebt_pred(vEBT_Leaf((Aa),(Uw)),aa_nat_nat(suc,zero_zero_nat)) = $ite((Aa),some_nat(zero_zero_nat),none_nat) ).

% vebt_pred.simps(2)
tff(fact_3096_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z)),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3097_divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xb),Z)),Y3) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xb),aa_real_real(times_times_real(Y3),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3098_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Y3) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(times_times_rat(Y3),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3099_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Y3),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),Y3)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3100_diff__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(Y3),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),Y3)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3101_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(Y3),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Y3)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3102_diff__frac__eq,axiom,
    ! [Y3: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,W),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W),Y3))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3103_diff__frac__eq,axiom,
    ! [Y3: real,Z: real,Xb: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xb),Y3)),aa_real_real(divide_divide_real(W),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Y3))),aa_real_real(times_times_real(Y3),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3104_diff__frac__eq,axiom,
    ! [Y3: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Y3)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y3))),aa_rat_rat(times_times_rat(Y3),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3105_add__divide__eq__if__simps_I4_J,axiom,
    ! [Aa: complex,B: complex,Z: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),Z)) = $ite(Z = zero_zero_complex,Aa,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3106_add__divide__eq__if__simps_I4_J,axiom,
    ! [Aa: real,B: real,Z: real] :
      aa_real_real(minus_minus_real(Aa),aa_real_real(divide_divide_real(B),Z)) = $ite(Z = zero_zero_real,Aa,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Aa),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3107_add__divide__eq__if__simps_I4_J,axiom,
    ! [Aa: rat,B: rat,Z: rat] :
      aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,Aa,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3108_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu),(B)),zero_zero_nat) = $ite((B),some_nat(one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_3109_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_3110_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_3111_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_3112_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_3113_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_le_numeral
tff(fact_3114_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_le_numeral
tff(fact_3115_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_3116_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_3117_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_3118_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_3119_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_3120_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_3121_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_less_numeral
tff(fact_3122_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_less_numeral
tff(fact_3123_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_3124_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_3125_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_3126_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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_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_3133_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_3134_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_3135_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_3136_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_3137_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3138_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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),Aa)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3139_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3140_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3141_zero__le__mult__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(Aa),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & 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),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_3142_zero__le__mult__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & 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),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_3143_zero__le__mult__iff,axiom,
    ! [Aa: 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,Aa),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          & 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),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_3144_mult__nonneg__nonpos2,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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(times_times_real(B),Aa)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3145_mult__nonneg__nonpos2,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( 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),Aa)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3146_mult__nonneg__nonpos2,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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),Aa)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3147_mult__nonneg__nonpos2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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),Aa)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3148_mult__nonpos__nonneg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(times_times_real(Aa),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_3149_mult__nonpos__nonneg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3150_mult__nonpos__nonneg,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3151_mult__nonpos__nonneg,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_3152_mult__nonneg__nonpos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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(times_times_real(Aa),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_3153_mult__nonneg__nonpos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( 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(Aa),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3154_mult__nonneg__nonpos,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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,Aa),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3155_mult__nonneg__nonpos,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_3156_mult__nonneg__nonneg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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(times_times_real(Aa),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3157_mult__nonneg__nonneg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( 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(Aa),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3158_mult__nonneg__nonneg,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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,Aa),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3159_mult__nonneg__nonneg,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3160_split__mult__neg__le,axiom,
    ! [Aa: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & 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),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(times_times_real(Aa),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_3161_split__mult__neg__le,axiom,
    ! [Aa: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & 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),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(Aa),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_3162_split__mult__neg__le,axiom,
    ! [Aa: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
          & 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),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,Aa),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_3163_split__mult__neg__le,axiom,
    ! [Aa: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          & 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),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,Aa),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_3164_mult__le__0__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & 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),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_3165_mult__le__0__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & 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),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_3166_mult__le__0__iff,axiom,
    ! [Aa: 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,Aa),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          & 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),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_3167_mult__right__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono
tff(fact_3168_mult__right__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono
tff(fact_3169_mult__right__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_3170_mult__right__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_3171_mult__right__mono__neg,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
     => ( 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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3172_mult__right__mono__neg,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( 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(Aa),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3173_mult__right__mono__neg,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3174_mult__left__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono
tff(fact_3175_mult__left__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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),Aa)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono
tff(fact_3176_mult__left__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_3177_mult__left__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_3178_mult__nonpos__nonpos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(times_times_real(Aa),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3179_mult__nonpos__nonpos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3180_mult__nonpos__nonpos,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3181_mult__left__mono__neg,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
     => ( 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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3182_mult__left__mono__neg,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( 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),Aa)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3183_mult__left__mono__neg,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
     => ( 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3184_split__mult__pos__le,axiom,
    ! [Aa: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & 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),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(times_times_real(Aa),B)) ) ).

% split_mult_pos_le
tff(fact_3185_split__mult__pos__le,axiom,
    ! [Aa: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & 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),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(Aa),B)) ) ).

% split_mult_pos_le
tff(fact_3186_split__mult__pos__le,axiom,
    ! [Aa: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          & 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),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,Aa),B)) ) ).

% split_mult_pos_le
tff(fact_3187_zero__le__square,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(Aa),Aa)) ).

% zero_le_square
tff(fact_3188_zero__le__square,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Aa)) ).

% zero_le_square
tff(fact_3189_zero__le__square,axiom,
    ! [Aa: 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,Aa),Aa)) ).

% zero_le_square
tff(fact_3190_mult__mono_H,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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),Aa)
         => ( 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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3191_mult__mono_H,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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),Aa)
         => ( 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(Aa),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3192_mult__mono_H,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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),Aa)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3193_mult__mono_H,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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),Aa)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3194_mult__mono,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3195_mult__mono,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3196_mult__mono,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3197_mult__mono,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3198_add__nonpos__eq__0__iff,axiom,
    ! [Xb: real,Y3: 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,Y3),zero_zero_real)
       => ( ( aa_real_real(plus_plus_real(Xb),Y3) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3199_add__nonpos__eq__0__iff,axiom,
    ! [Xb: rat,Y3: 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,Y3),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Y3) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3200_add__nonpos__eq__0__iff,axiom,
    ! [Xb: nat,Y3: 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,Y3),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y3) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3201_add__nonpos__eq__0__iff,axiom,
    ! [Xb: int,Y3: 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,Y3),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y3) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3202_add__nonneg__eq__0__iff,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( ( aa_real_real(plus_plus_real(Xb),Y3) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3203_add__nonneg__eq__0__iff,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Y3) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3204_add__nonneg__eq__0__iff,axiom,
    ! [Xb: nat,Y3: 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),Y3)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y3) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3205_add__nonneg__eq__0__iff,axiom,
    ! [Xb: int,Y3: 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),Y3)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y3) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3206_add__nonpos__nonpos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(plus_plus_real(Aa),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_3207_add__nonpos__nonpos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_3208_add__nonpos__nonpos,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_3209_add__nonpos__nonpos,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_3210_add__nonneg__nonneg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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(plus_plus_real(Aa),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3211_add__nonneg__nonneg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( 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(Aa),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3212_add__nonneg__nonneg,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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,Aa),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3213_add__nonneg__nonneg,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3214_add__increasing2,axiom,
    ! [C: real,B: real,Aa: 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),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(plus_plus_real(Aa),C)) ) ) ).

% add_increasing2
tff(fact_3215_add__increasing2,axiom,
    ! [C: rat,B: rat,Aa: 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),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(Aa),C)) ) ) ).

% add_increasing2
tff(fact_3216_add__increasing2,axiom,
    ! [C: nat,B: nat,Aa: 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),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)) ) ) ).

% add_increasing2
tff(fact_3217_add__increasing2,axiom,
    ! [C: int,B: int,Aa: 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),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)) ) ) ).

% add_increasing2
tff(fact_3218_add__decreasing2,axiom,
    ! [C: real,Aa: 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,Aa),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Aa),C)),B) ) ) ).

% add_decreasing2
tff(fact_3219_add__decreasing2,axiom,
    ! [C: rat,Aa: 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,Aa),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),C)),B) ) ) ).

% add_decreasing2
tff(fact_3220_add__decreasing2,axiom,
    ! [C: nat,Aa: 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,Aa),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),B) ) ) ).

% add_decreasing2
tff(fact_3221_add__decreasing2,axiom,
    ! [C: int,Aa: 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,Aa),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),B) ) ) ).

% add_decreasing2
tff(fact_3222_add__increasing,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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(plus_plus_real(Aa),C)) ) ) ).

% add_increasing
tff(fact_3223_add__increasing,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( 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(Aa),C)) ) ) ).

% add_increasing
tff(fact_3224_add__increasing,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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,Aa),C)) ) ) ).

% add_increasing
tff(fact_3225_add__increasing,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),C)) ) ) ).

% add_increasing
tff(fact_3226_add__decreasing,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(plus_plus_real(Aa),C)),B) ) ) ).

% add_decreasing
tff(fact_3227_add__decreasing,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),C)),B) ) ) ).

% add_decreasing
tff(fact_3228_add__decreasing,axiom,
    ! [Aa: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),C)),B) ) ) ).

% add_decreasing
tff(fact_3229_add__decreasing,axiom,
    ! [Aa: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),C)),B) ) ) ).

% add_decreasing
tff(fact_3230_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_3231_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_3232_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_3233_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_3234_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_3235_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_3236_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_3237_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_3238_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_3239_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_3240_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_3241_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_3242_mult__neg__neg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(times_times_real(Aa),B)) ) ) ).

% mult_neg_neg
tff(fact_3243_mult__neg__neg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(Aa),B)) ) ) ).

% mult_neg_neg
tff(fact_3244_mult__neg__neg,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),B)) ) ) ).

% mult_neg_neg
tff(fact_3245_not__square__less__zero,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),Aa)),zero_zero_real) ).

% not_square_less_zero
tff(fact_3246_not__square__less__zero,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),Aa)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_3247_not__square__less__zero,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Aa)),zero_zero_int) ).

% not_square_less_zero
tff(fact_3248_mult__less__0__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & 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),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3249_mult__less__0__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & 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),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3250_mult__less__0__iff,axiom,
    ! [Aa: 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,Aa),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
          & 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),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3251_mult__neg__pos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(times_times_real(Aa),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_3252_mult__neg__pos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(Aa),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_3253_mult__neg__pos,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_3254_mult__neg__pos,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_3255_mult__pos__neg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( 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(times_times_real(Aa),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_3256_mult__pos__neg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( 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(Aa),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_3257_mult__pos__neg,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( 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,Aa),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_3258_mult__pos__neg,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( 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,Aa),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_3259_mult__pos__pos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( 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(times_times_real(Aa),B)) ) ) ).

% mult_pos_pos
tff(fact_3260_mult__pos__pos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( 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(Aa),B)) ) ) ).

% mult_pos_pos
tff(fact_3261_mult__pos__pos,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( 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,Aa),B)) ) ) ).

% mult_pos_pos
tff(fact_3262_mult__pos__pos,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( 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,Aa),B)) ) ) ).

% mult_pos_pos
tff(fact_3263_mult__pos__neg2,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( 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(times_times_real(B),Aa)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_3264_mult__pos__neg2,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( 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),Aa)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_3265_mult__pos__neg2,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( 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),Aa)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_3266_mult__pos__neg2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( 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),Aa)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_3267_zero__less__mult__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(Aa),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & 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),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_3268_zero__less__mult__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & 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),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_3269_zero__less__mult__iff,axiom,
    ! [Aa: 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,Aa),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
          & 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),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_3270_zero__less__mult__pos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(Aa),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_3271_zero__less__mult__pos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_3272_zero__less__mult__pos,axiom,
    ! [Aa: 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,Aa),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_3273_zero__less__mult__pos,axiom,
    ! [Aa: 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,Aa),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_3274_zero__less__mult__pos2,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(B),Aa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3275_zero__less__mult__pos2,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),Aa))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3276_zero__less__mult__pos2,axiom,
    ! [B: nat,Aa: 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),Aa))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3277_zero__less__mult__pos2,axiom,
    ! [B: int,Aa: 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),Aa))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3278_mult__less__cancel__left__neg,axiom,
    ! [C: real,Aa: 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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3279_mult__less__cancel__left__neg,axiom,
    ! [C: rat,Aa: 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),Aa)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3280_mult__less__cancel__left__neg,axiom,
    ! [C: int,Aa: 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),Aa)),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),Aa) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3281_mult__less__cancel__left__pos,axiom,
    ! [C: real,Aa: 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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3282_mult__less__cancel__left__pos,axiom,
    ! [C: rat,Aa: 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),Aa)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3283_mult__less__cancel__left__pos,axiom,
    ! [C: int,Aa: 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3284_mult__strict__left__mono__neg,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ( 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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3285_mult__strict__left__mono__neg,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ( 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),Aa)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3286_mult__strict__left__mono__neg,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => ( 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3287_mult__strict__left__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3288_mult__strict__left__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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),Aa)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3289_mult__strict__left__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3290_mult__strict__left__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3291_mult__less__cancel__left__disj,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),Aa)),aa_real_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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3292_mult__less__cancel__left__disj,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3293_mult__less__cancel__left__disj,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3294_mult__strict__right__mono__neg,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ( 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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3295_mult__strict__right__mono__neg,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ( 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(Aa),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3296_mult__strict__right__mono__neg,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3297_mult__strict__right__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3298_mult__strict__right__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(Aa),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3299_mult__strict__right__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3300_mult__strict__right__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3301_mult__less__cancel__right__disj,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),C)),aa_real_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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3302_mult__less__cancel__right__disj,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3303_mult__less__cancel__right__disj,axiom,
    ! [Aa: 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,Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3304_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3305_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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),Aa)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3306_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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),Aa)),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_3307_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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),Aa)),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_3308_add__less__zeroD,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Xb),Y3)),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,Y3),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_3309_add__less__zeroD,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xb),Y3)),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,Y3),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_3310_add__less__zeroD,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y3)),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,Y3),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_3311_pos__add__strict,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( 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(plus_plus_real(Aa),C)) ) ) ).

% pos_add_strict
tff(fact_3312_pos__add__strict,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( 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(Aa),C)) ) ) ).

% pos_add_strict
tff(fact_3313_pos__add__strict,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( 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,Aa),C)) ) ) ).

% pos_add_strict
tff(fact_3314_pos__add__strict,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( 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,Aa),C)) ) ) ).

% pos_add_strict
tff(fact_3315_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_3316_add__pos__pos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( 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(plus_plus_real(Aa),B)) ) ) ).

% add_pos_pos
tff(fact_3317_add__pos__pos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( 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(Aa),B)) ) ) ).

% add_pos_pos
tff(fact_3318_add__pos__pos,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( 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,Aa),B)) ) ) ).

% add_pos_pos
tff(fact_3319_add__pos__pos,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( 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,Aa),B)) ) ) ).

% add_pos_pos
tff(fact_3320_add__neg__neg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(plus_plus_real(Aa),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_3321_add__neg__neg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(Aa),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_3322_add__neg__neg,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_3323_add__neg__neg,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_3324_le__iff__diff__le__0,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Aa),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_3325_le__iff__diff__le__0,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Aa),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_3326_le__iff__diff__le__0,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_3327_less__iff__diff__less__0,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Aa),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_3328_less__iff__diff__less__0,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Aa),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_3329_less__iff__diff__less__0,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_3330_zero__le__power,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ) ).

% zero_le_power
tff(fact_3331_zero__le__power,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),Nb)) ) ).

% zero_le_power
tff(fact_3332_zero__le__power,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => 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,Aa),Nb)) ) ).

% zero_le_power
tff(fact_3333_zero__le__power,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ) ).

% zero_le_power
tff(fact_3334_power__mono,axiom,
    ! [Aa: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono
tff(fact_3335_power__mono,axiom,
    ! [Aa: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_3336_power__mono,axiom,
    ! [Aa: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_3337_power__mono,axiom,
    ! [Aa: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono
tff(fact_3338_zero__less__power,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => 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),Nb)) ) ).

% zero_less_power
tff(fact_3339_zero__less__power,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),Nb)) ) ).

% zero_less_power
tff(fact_3340_zero__less__power,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => 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),Nb)) ) ).

% zero_less_power
tff(fact_3341_zero__less__power,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => 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),Nb)) ) ).

% zero_less_power
tff(fact_3342_power__0,axiom,
    ! [Aa: rat] : aa_nat_rat(power_power_rat(Aa),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_3343_power__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_3344_power__0,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_3345_power__0,axiom,
    ! [Aa: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_3346_power__0,axiom,
    ! [Aa: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_3347_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_3348_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_3349_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_3350_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( ( Mb = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Mb = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_3351_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_3352_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I5)) ) ) ) ).

% All_less_Suc2
tff(fact_3353_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M5: nat] : Nb = aa_nat_nat(suc,M5) ) ).

% gr0_conv_Suc
tff(fact_3354_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I5)) ) ) ) ).

% Ex_less_Suc2
tff(fact_3355_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_3356_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_3357_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_3358_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_3359_option_Osize_I4_J,axiom,
    ! [X23: nat] : size_size_option_nat(some_nat(X23)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3360_option_Osize_I4_J,axiom,
    ! [X23: num] : size_size_option_num(some_num(X23)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3361_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_3362_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_3363_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,
    ! [Aa: $o,B: $o,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((Aa),(B))),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_3364_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3365_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3366_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3367_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_3368_nat__mult__less__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_3369_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_3370_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_3371_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_3372_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_3373_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = zero_zero_nat ).

% diff_add_0
tff(fact_3374_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Mb = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3375_nat__power__less__imp__less,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_3376_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts2,S),Xb) = vEBT_Node(Info,zero_zero_nat,Ts2,S) ).

% vebt_insert.simps(2)
tff(fact_3377_vebt__pred_Osimps_I3_J,axiom,
    ! [Aa: $o,B: $o,Va: nat] :
      vEBT_vebt_pred(vEBT_Leaf((Aa),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((Aa),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(3)
tff(fact_3378_field__le__mult__one__interval,axiom,
    ! [Xb: real,Y3: 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(times_times_real(Z3),Xb)),Y3) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ).

% field_le_mult_one_interval
tff(fact_3379_field__le__mult__one__interval,axiom,
    ! [Xb: rat,Y3: 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)),Y3) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3) ) ).

% field_le_mult_one_interval
tff(fact_3380_le__divide__eq__1,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),Aa))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa) ) ) ) ).

% le_divide_eq_1
tff(fact_3381_le__divide__eq__1,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),Aa))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ) ) ).

% le_divide_eq_1
tff(fact_3382_divide__le__eq__1,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),Aa)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) )
        | ( Aa = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3383_divide__le__eq__1,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),Aa)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) )
        | ( Aa = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3384_divide__left__mono__neg,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(times_times_real(Aa),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),Aa)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3385_divide__left__mono__neg,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),Aa)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3386_mult__imp__le__div__pos,axiom,
    ! [Y3: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),Y3)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(divide_divide_real(Xb),Y3)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3387_mult__imp__le__div__pos,axiom,
    ! [Y3: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y3)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Y3)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3388_mult__imp__div__pos__le,axiom,
    ! [Y3: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(times_times_real(Z),Y3))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y3)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3389_mult__imp__div__pos__le,axiom,
    ! [Y3: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(times_times_rat(Z),Y3))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3390_pos__le__divide__eq,axiom,
    ! [C: real,Aa: 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),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3391_pos__le__divide__eq,axiom,
    ! [C: rat,Aa: 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),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3392_pos__divide__le__eq,axiom,
    ! [C: real,B: real,Aa: 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(divide_divide_real(B),C)),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(Aa),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3393_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,Aa: 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(divide_divide_rat(B),C)),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3394_neg__le__divide__eq,axiom,
    ! [C: real,Aa: 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),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(Aa),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3395_neg__le__divide__eq,axiom,
    ! [C: rat,Aa: 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),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3396_neg__divide__le__eq,axiom,
    ! [C: real,B: real,Aa: 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(divide_divide_real(B),C)),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3397_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,Aa: 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(divide_divide_rat(B),C)),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3398_divide__left__mono,axiom,
    ! [B: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa)
     => ( 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(times_times_real(Aa),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),Aa)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3399_divide__left__mono,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa)
     => ( 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(Aa),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),Aa)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3400_le__divide__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_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(times_times_real(Aa),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(times_times_real(Aa),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_3401_le__divide__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),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(Aa),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3402_divide__le__eq,axiom,
    ! [B: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),Aa)
    <=> $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(times_times_real(Aa),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(times_times_real(Aa),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)) ) ) ).

% divide_le_eq
tff(fact_3403_divide__le__eq,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),Aa)
    <=> $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(Aa),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(Aa),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)) ) ) ).

% divide_le_eq
tff(fact_3404_frac__le__eq,axiom,
    ! [Y3: real,Z: real,Xb: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y3)),aa_real_real(divide_divide_real(W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Y3))),aa_real_real(times_times_real(Y3),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_3405_frac__le__eq,axiom,
    ! [Y3: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y3))),aa_rat_rat(times_times_rat(Y3),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_3406_frac__less__eq,axiom,
    ! [Y3: real,Z: real,Xb: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Y3)),aa_real_real(divide_divide_real(W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Y3))),aa_real_real(times_times_real(Y3),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_3407_frac__less__eq,axiom,
    ! [Y3: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y3))),aa_rat_rat(times_times_rat(Y3),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_3408_xor__num_Ocases,axiom,
    ! [Xb: product_prod_num_num] :
      ( ( Xb != product_Pair_num_num(one,one) )
     => ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),one)
           => ( ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit0,N))
             => ( ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit1,N))
               => ( ! [M2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),one)
                 => ( ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit0,N))
                   => ~ ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_3409_Diff__single__insert,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),B2)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(Xb,B2)) ) ).

% Diff_single_insert
tff(fact_3410_Diff__single__insert,axiom,
    ! [A: set_real,Xb: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A,insert_real(Xb,bot_bot_set_real))),B2)
     => aa_set_real_o(ord_less_eq_set_real(A),insert_real(Xb,B2)) ) ).

% Diff_single_insert
tff(fact_3411_Diff__single__insert,axiom,
    ! [A: set_nat,Xb: nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A,insert_nat(Xb,bot_bot_set_nat))),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(Xb,B2)) ) ).

% Diff_single_insert
tff(fact_3412_Diff__single__insert,axiom,
    ! [A: set_int,Xb: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A,insert_int(Xb,bot_bot_set_int))),B2)
     => aa_set_int_o(ord_less_eq_set_int(A),insert_int(Xb,B2)) ) ).

% Diff_single_insert
tff(fact_3413_subset__insert__iff,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(Xb,B2))
    <=> $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),B2),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B2)) ) ).

% subset_insert_iff
tff(fact_3414_subset__insert__iff,axiom,
    ! [A: set_complex,Xb: complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),insert_complex(Xb,B2))
    <=> $ite(aa_set_complex_o(member_complex(Xb),A),aa_set_complex_o(ord_le211207098394363844omplex(minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex))),B2),aa_set_complex_o(ord_le211207098394363844omplex(A),B2)) ) ).

% subset_insert_iff
tff(fact_3415_subset__insert__iff,axiom,
    ! [A: set_set_nat,Xb: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),insert_set_nat(Xb,B2))
    <=> $ite(aa_set_set_nat_o(member_set_nat(Xb),A),aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A,insert_set_nat(Xb,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2)) ) ).

% subset_insert_iff
tff(fact_3416_subset__insert__iff,axiom,
    ! [A: set_real,Xb: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),insert_real(Xb,B2))
    <=> $ite(aa_set_real_o(member_real(Xb),A),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A,insert_real(Xb,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A),B2)) ) ).

% subset_insert_iff
tff(fact_3417_subset__insert__iff,axiom,
    ! [A: set_nat,Xb: nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(Xb,B2))
    <=> $ite(aa_set_nat_o(member_nat(Xb),A),aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A,insert_nat(Xb,bot_bot_set_nat))),B2),aa_set_nat_o(ord_less_eq_set_nat(A),B2)) ) ).

% subset_insert_iff
tff(fact_3418_subset__insert__iff,axiom,
    ! [A: set_int,Xb: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),insert_int(Xb,B2))
    <=> $ite(aa_set_int_o(member_int(Xb),A),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A,insert_int(Xb,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A),B2)) ) ).

% subset_insert_iff
tff(fact_3419_psubset__insert__iff,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A),insert_VEBT_VEBT(Xb,B2))
    <=> $ite(
          aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),B2),
          aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A),B2),
          $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A),aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),B2),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B2)) ) ) ).

% psubset_insert_iff
tff(fact_3420_psubset__insert__iff,axiom,
    ! [A: set_complex,Xb: complex,B2: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A),insert_complex(Xb,B2))
    <=> $ite(
          aa_set_complex_o(member_complex(Xb),B2),
          aa_set_complex_o(ord_less_set_complex(A),B2),
          $ite(aa_set_complex_o(member_complex(Xb),A),aa_set_complex_o(ord_less_set_complex(minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex))),B2),aa_set_complex_o(ord_le211207098394363844omplex(A),B2)) ) ) ).

% psubset_insert_iff
tff(fact_3421_psubset__insert__iff,axiom,
    ! [A: set_set_nat,Xb: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A),insert_set_nat(Xb,B2))
    <=> $ite(
          aa_set_set_nat_o(member_set_nat(Xb),B2),
          aa_set_set_nat_o(ord_less_set_set_nat(A),B2),
          $ite(aa_set_set_nat_o(member_set_nat(Xb),A),aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(A,insert_set_nat(Xb,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),B2)) ) ) ).

% psubset_insert_iff
tff(fact_3422_psubset__insert__iff,axiom,
    ! [A: set_real,Xb: real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A),insert_real(Xb,B2))
    <=> $ite(
          aa_set_real_o(member_real(Xb),B2),
          aa_set_real_o(ord_less_set_real(A),B2),
          $ite(aa_set_real_o(member_real(Xb),A),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A,insert_real(Xb,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A),B2)) ) ) ).

% psubset_insert_iff
tff(fact_3423_psubset__insert__iff,axiom,
    ! [A: set_nat,Xb: nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),insert_nat(Xb,B2))
    <=> $ite(
          aa_set_nat_o(member_nat(Xb),B2),
          aa_set_nat_o(ord_less_set_nat(A),B2),
          $ite(aa_set_nat_o(member_nat(Xb),A),aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(A,insert_nat(Xb,bot_bot_set_nat))),B2),aa_set_nat_o(ord_less_eq_set_nat(A),B2)) ) ) ).

% psubset_insert_iff
tff(fact_3424_psubset__insert__iff,axiom,
    ! [A: set_int,Xb: int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),insert_int(Xb,B2))
    <=> $ite(
          aa_set_int_o(member_int(Xb),B2),
          aa_set_int_o(ord_less_set_int(A),B2),
          $ite(aa_set_int_o(member_int(Xb),A),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A,insert_int(Xb,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A),B2)) ) ) ).

% psubset_insert_iff
tff(fact_3425_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 != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_3426_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,
    ! [Aa: $o,B: $o,Nb: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Leaf((Aa),(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_3427_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [Aa: $o,B: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Leaf((Aa),(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_3428_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,
    ! [Aa: $o,B: $o,Xb: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Leaf((Aa),(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_3429_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3430_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ).

% linordered_field_no_lb
tff(fact_3431_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ).

% linordered_field_no_lb
tff(fact_3432_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_12: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X_12) ).

% linordered_field_no_ub
tff(fact_3433_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_12: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),X_12) ).

% linordered_field_no_ub
tff(fact_3434_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_3435_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_3436_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_3437_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_3438_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3439_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,Nb)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_3440_mult__le__cancel__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),Aa)),aa_real_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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_left
tff(fact_3441_mult__le__cancel__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_left
tff(fact_3442_mult__le__cancel__left,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_left
tff(fact_3443_mult__le__cancel__right,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),C)),aa_real_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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_right
tff(fact_3444_mult__le__cancel__right,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_right
tff(fact_3445_mult__le__cancel__right,axiom,
    ! [Aa: 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,Aa),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_right
tff(fact_3446_mult__left__less__imp__less,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),Aa)),aa_real_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,Aa),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3447_mult__left__less__imp__less,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),Aa)),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,Aa),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3448_mult__left__less__imp__less,axiom,
    ! [C: nat,Aa: 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),Aa)),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,Aa),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3449_mult__left__less__imp__less,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3450_mult__strict__mono,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3451_mult__strict__mono,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(Aa),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3452_mult__strict__mono,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3453_mult__strict__mono,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3454_mult__less__cancel__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),Aa)),aa_real_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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left
tff(fact_3455_mult__less__cancel__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left
tff(fact_3456_mult__less__cancel__left,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left
tff(fact_3457_mult__right__less__imp__less,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),C)),aa_real_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,Aa),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3458_mult__right__less__imp__less,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),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,Aa),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3459_mult__right__less__imp__less,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3460_mult__right__less__imp__less,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3461_mult__strict__mono_H,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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),Aa)
         => ( 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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3462_mult__strict__mono_H,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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),Aa)
         => ( 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(Aa),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3463_mult__strict__mono_H,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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),Aa)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3464_mult__strict__mono_H,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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),Aa)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3465_mult__less__cancel__right,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),C)),aa_real_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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right
tff(fact_3466_mult__less__cancel__right,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right
tff(fact_3467_mult__less__cancel__right,axiom,
    ! [Aa: 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,Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right
tff(fact_3468_mult__le__cancel__left__neg,axiom,
    ! [C: real,Aa: 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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),Aa) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3469_mult__le__cancel__left__neg,axiom,
    ! [C: rat,Aa: 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),Aa)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3470_mult__le__cancel__left__neg,axiom,
    ! [C: int,Aa: 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),Aa)),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),Aa) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3471_mult__le__cancel__left__pos,axiom,
    ! [C: real,Aa: 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(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3472_mult__le__cancel__left__pos,axiom,
    ! [C: rat,Aa: 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),Aa)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3473_mult__le__cancel__left__pos,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3474_mult__left__le__imp__le,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),Aa)),aa_real_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,Aa),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3475_mult__left__le__imp__le,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),Aa)),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,Aa),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3476_mult__left__le__imp__le,axiom,
    ! [C: nat,Aa: 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),Aa)),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,Aa),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3477_mult__left__le__imp__le,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3478_mult__right__le__imp__le,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),C)),aa_real_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,Aa),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3479_mult__right__le__imp__le,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),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,Aa),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3480_mult__right__le__imp__le,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3481_mult__right__le__imp__le,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3482_mult__le__less__imp__less,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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),Aa)
         => ( 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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3483_mult__le__less__imp__less,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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),Aa)
         => ( 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(Aa),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3484_mult__le__less__imp__less,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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),Aa)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3485_mult__le__less__imp__less,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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),Aa)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3486_mult__less__le__imp__less,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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),Aa)
         => ( 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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3487_mult__less__le__imp__less,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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),Aa)
         => ( 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(Aa),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3488_mult__less__le__imp__less,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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),Aa)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3489_mult__less__le__imp__less,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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),Aa)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3490_add__strict__increasing2,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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(plus_plus_real(Aa),C)) ) ) ).

% add_strict_increasing2
tff(fact_3491_add__strict__increasing2,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( 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(Aa),C)) ) ) ).

% add_strict_increasing2
tff(fact_3492_add__strict__increasing2,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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,Aa),C)) ) ) ).

% add_strict_increasing2
tff(fact_3493_add__strict__increasing2,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),C)) ) ) ).

% add_strict_increasing2
tff(fact_3494_add__strict__increasing,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( 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(plus_plus_real(Aa),C)) ) ) ).

% add_strict_increasing
tff(fact_3495_add__strict__increasing,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( 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(Aa),C)) ) ) ).

% add_strict_increasing
tff(fact_3496_add__strict__increasing,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( 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,Aa),C)) ) ) ).

% add_strict_increasing
tff(fact_3497_add__strict__increasing,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( 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,Aa),C)) ) ) ).

% add_strict_increasing
tff(fact_3498_add__pos__nonneg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( 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(plus_plus_real(Aa),B)) ) ) ).

% add_pos_nonneg
tff(fact_3499_add__pos__nonneg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( 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(Aa),B)) ) ) ).

% add_pos_nonneg
tff(fact_3500_add__pos__nonneg,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( 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,Aa),B)) ) ) ).

% add_pos_nonneg
tff(fact_3501_add__pos__nonneg,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( 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,Aa),B)) ) ) ).

% add_pos_nonneg
tff(fact_3502_add__nonpos__neg,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(plus_plus_real(Aa),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_3503_add__nonpos__neg,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_3504_add__nonpos__neg,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_3505_add__nonpos__neg,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_3506_add__nonneg__pos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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(plus_plus_real(Aa),B)) ) ) ).

% add_nonneg_pos
tff(fact_3507_add__nonneg__pos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( 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(Aa),B)) ) ) ).

% add_nonneg_pos
tff(fact_3508_add__nonneg__pos,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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,Aa),B)) ) ) ).

% add_nonneg_pos
tff(fact_3509_add__nonneg__pos,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),B)) ) ) ).

% add_nonneg_pos
tff(fact_3510_add__neg__nonpos,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(plus_plus_real(Aa),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_3511_add__neg__nonpos,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),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(Aa),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_3512_add__neg__nonpos,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_3513_add__neg__nonpos,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_3514_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [Aa: $o,B: $o,Va: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf((Aa),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_3515_div__positive,axiom,
    ! [B: nat,Aa: 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),Aa)
       => 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),B)) ) ) ).

% div_positive
tff(fact_3516_div__positive,axiom,
    ! [B: int,Aa: 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),Aa)
       => 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,Aa),B)) ) ) ).

% div_positive
tff(fact_3517_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3518_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3519_mult__left__le__one__le,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Y3),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_3520_mult__left__le__one__le,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y3),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_3521_mult__left__le__one__le,axiom,
    ! [Xb: int,Y3: 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),Y3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),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,Y3),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_3522_mult__right__le__one__le,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Xb),Y3)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_3523_mult__right__le__one__le,axiom,
    ! [Xb: rat,Y3: 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),Y3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Y3)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_3524_mult__right__le__one__le,axiom,
    ! [Xb: int,Y3: 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),Y3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),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),Y3)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_3525_mult__le__one,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(times_times_real(Aa),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_3526_mult__le__one,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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(Aa),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_3527_mult__le__one,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_3528_mult__le__one,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_3529_mult__left__le,axiom,
    ! [C: real,Aa: 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),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),C)),Aa) ) ) ).

% mult_left_le
tff(fact_3530_mult__left__le,axiom,
    ! [C: rat,Aa: 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),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),Aa) ) ) ).

% mult_left_le
tff(fact_3531_mult__left__le,axiom,
    ! [C: nat,Aa: 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),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),Aa) ) ) ).

% mult_left_le
tff(fact_3532_mult__left__le,axiom,
    ! [C: int,Aa: 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),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),Aa) ) ) ).

% mult_left_le
tff(fact_3533_sum__squares__ge__zero,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y3),Y3))) ).

% sum_squares_ge_zero
tff(fact_3534_sum__squares__ge__zero,axiom,
    ! [Xb: rat,Y3: 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(Y3),Y3))) ).

% sum_squares_ge_zero
tff(fact_3535_sum__squares__ge__zero,axiom,
    ! [Xb: int,Y3: 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,Y3),Y3))) ).

% sum_squares_ge_zero
tff(fact_3536_sum__squares__le__zero__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y3),Y3))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3537_sum__squares__le__zero__iff,axiom,
    ! [Xb: rat,Y3: 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(Y3),Y3))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3538_sum__squares__le__zero__iff,axiom,
    ! [Xb: int,Y3: 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,Y3),Y3))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3539_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_3540_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_3541_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_3542_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_3543_not__sum__squares__lt__zero,axiom,
    ! [Xb: real,Y3: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y3),Y3))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_3544_not__sum__squares__lt__zero,axiom,
    ! [Xb: rat,Y3: 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(Y3),Y3))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_3545_not__sum__squares__lt__zero,axiom,
    ! [Xb: int,Y3: 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,Y3),Y3))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_3546_sum__squares__gt__zero__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y3),Y3)))
    <=> ( ( Xb != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3547_sum__squares__gt__zero__iff,axiom,
    ! [Xb: rat,Y3: 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(Y3),Y3)))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3548_sum__squares__gt__zero__iff,axiom,
    ! [Xb: int,Y3: 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,Y3),Y3)))
    <=> ( ( Xb != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3549_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(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_3550_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(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_3551_power__less__imp__less__base,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% power_less_imp_less_base
tff(fact_3552_power__less__imp__less__base,axiom,
    ! [Aa: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),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,Aa),B) ) ) ).

% power_less_imp_less_base
tff(fact_3553_power__less__imp__less__base,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% power_less_imp_less_base
tff(fact_3554_power__less__imp__less__base,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% power_less_imp_less_base
tff(fact_3555_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,Aa: 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,Aa),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,Aa),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3556_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,Aa: 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,Aa),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,Aa),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3557_power__le__one,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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,Aa),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_3558_power__le__one,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_3559_power__le__one,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_3560_power__le__one,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_3561_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3562_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C) = B,aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3563_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) = B,aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3564_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3565_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = aa_num_real(numeral_numeral_real,W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C),aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3566_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C),aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3567_div__add__self2,axiom,
    ! [B: nat,Aa: 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),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_3568_div__add__self2,axiom,
    ! [B: int,Aa: 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),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3569_div__add__self1,axiom,
    ! [B: nat,Aa: 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),Aa)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_3570_div__add__self1,axiom,
    ! [B: int,Aa: 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),Aa)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3571_power__le__imp__le__base,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% power_le_imp_le_base
tff(fact_3572_power__le__imp__le__base,axiom,
    ! [Aa: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),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,Aa),B) ) ) ).

% power_le_imp_le_base
tff(fact_3573_power__le__imp__le__base,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% power_le_imp_le_base
tff(fact_3574_power__le__imp__le__base,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% power_le_imp_le_base
tff(fact_3575_power__inject__base,axiom,
    ! [Aa: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( Aa = B ) ) ) ) ).

% power_inject_base
tff(fact_3576_power__inject__base,axiom,
    ! [Aa: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(Aa),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),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( Aa = B ) ) ) ) ).

% power_inject_base
tff(fact_3577_power__inject__base,axiom,
    ! [Aa: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),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),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( Aa = B ) ) ) ) ).

% power_inject_base
tff(fact_3578_power__inject__base,axiom,
    ! [Aa: int,Nb: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),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),Aa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( Aa = B ) ) ) ) ).

% power_inject_base
tff(fact_3579_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_3580_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_3581_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_3582_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)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_3583_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_3584_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_3585_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_3586_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_3587_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_3588_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_3589_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Mb)) ) ) ).

% div_le_mono2
tff(fact_3590_length__pos__if__in__set,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3591_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_3592_length__pos__if__in__set,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xb),set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3593_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_3594_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_3595_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_3596_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_3597_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_3598_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Mb )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_3599_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,Aa: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D4: nat] :
              ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
              & ~ aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_3600_nat__diff__split,axiom,
    ! [P: fun_nat_o,Aa: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D4: nat] :
            ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
           => aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split
tff(fact_3601_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_3602_nat__mult__le__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb)),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,Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_3603_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_3604_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Mb: 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,Mb),Q2)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_3605_nat__mult__div__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_3606_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_3607_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S),Xb) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S) ).

% vebt_insert.simps(3)
tff(fact_3608_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xb) = Y3 )
     => ( ! [A2: $o] :
            ( ? [B3: $o] : Xb = vEBT_Leaf((A2),(B3))
           => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((A2),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 = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y3 != one_one_nat ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2,Uz2)
             => ( Y3 != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_3609_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: 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),S)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(U),aa_real_real(divide_divide_real(aa_real_real(times_times_real(R2),aa_real_real(minus_minus_real(V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_3610_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: 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),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_3611_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero_nat,Ts2,S)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_3612_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero_nat,Ts2,S)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_3613_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A2: $o,B3: $o] : Xb != vEBT_Leaf((A2),(B3))
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != 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_3614_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Aa: $o,B: $o,Va: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((Aa),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = 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_3615_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y3: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y3) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y3) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Y3) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y3) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ (Y3) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => (Y3) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3616_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y3 )
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => ( Y3 != $ite(
                  (A2),
                  some_nat(zero_zero_nat),
                  $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y3 != none_nat ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y3 != some_nat(Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_3617_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y3 )
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => ( Y3 != $ite(
                  (B3),
                  some_nat(one_one_nat),
                  $ite((A2),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y3 != none_nat ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y3 != some_nat(Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_3618_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(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_3619_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_3620_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_3621_mult__le__cancel__left2,axiom,
    ! [C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3622_mult__le__cancel__left2,axiom,
    ! [C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3623_mult__le__cancel__left2,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3624_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(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_3625_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_3626_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_3627_mult__le__cancel__right2,axiom,
    ! [Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3628_mult__le__cancel__right2,axiom,
    ! [Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3629_mult__le__cancel__right2,axiom,
    ! [Aa: 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,Aa),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3630_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_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_3631_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_3632_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_3633_mult__less__cancel__left2,axiom,
    ! [C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3634_mult__less__cancel__left2,axiom,
    ! [C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3635_mult__less__cancel__left2,axiom,
    ! [C: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3636_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_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_3637_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_3638_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_3639_mult__less__cancel__right2,axiom,
    ! [Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3640_mult__less__cancel__right2,axiom,
    ! [Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3641_mult__less__cancel__right2,axiom,
    ! [Aa: 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,Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3642_convex__bound__le,axiom,
    ! [Xb: real,Aa: real,Y3: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Aa)
       => ( 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(plus_plus_real(U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),Xb)),aa_real_real(times_times_real(V),Y3))),Aa) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3643_convex__bound__le,axiom,
    ! [Xb: rat,Aa: rat,Y3: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Aa)
       => ( 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),Y3))),Aa) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3644_convex__bound__le,axiom,
    ! [Xb: int,Aa: int,Y3: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Aa)
       => ( 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),Y3))),Aa) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3645_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_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(times_times_real(aa_num_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(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3646_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_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(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3647_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(divide_divide_real(B),C)),aa_num_real(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(times_times_real(aa_num_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(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3648_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_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(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3649_power__Suc__less,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ) ) ).

% power_Suc_less
tff(fact_3650_power__Suc__less,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(Aa),Nb))),aa_nat_rat(power_power_rat(Aa),Nb)) ) ) ).

% power_Suc_less
tff(fact_3651_power__Suc__less,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) ) ) ).

% power_Suc_less
tff(fact_3652_power__Suc__less,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ) ) ).

% power_Suc_less
tff(fact_3653_power__Suc__le__self,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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,Aa),aa_nat_nat(suc,Nb))),Aa) ) ) ).

% power_Suc_le_self
tff(fact_3654_power__Suc__le__self,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Nb))),Aa) ) ) ).

% power_Suc_le_self
tff(fact_3655_power__Suc__le__self,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),aa_nat_nat(suc,Nb))),Aa) ) ) ).

% power_Suc_le_self
tff(fact_3656_power__Suc__le__self,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),aa_nat_nat(suc,Nb))),Aa) ) ) ).

% power_Suc_le_self
tff(fact_3657_power__Suc__less__one,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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,Aa),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_3658_power__Suc__less__one,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_3659_power__Suc__less__one,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_3660_power__Suc__less__one,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_3661_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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,Aa),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3662_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),N3)),aa_nat_rat(power_power_rat(Aa),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3663_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3664_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),N3)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3665_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_3666_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_3667_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_3668_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_3669_zero__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_3670_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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,Aa),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3671_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,Aa: rat] :
      ( 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,zero_zero_rat),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),N3)),aa_nat_rat(power_power_rat(Aa),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3672_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,Aa: nat] :
      ( 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,zero_zero_nat),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3673_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,Aa: int] :
      ( 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,zero_zero_int),Aa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),N3)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3674_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_3675_self__le__power,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Aa)
     => ( 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),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ) ) ).

% self_le_power
tff(fact_3676_self__le__power,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Aa)
     => ( 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),aa_nat_rat(power_power_rat(Aa),Nb)) ) ) ).

% self_le_power
tff(fact_3677_self__le__power,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Aa)
     => ( 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),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) ) ) ).

% self_le_power
tff(fact_3678_self__le__power,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Aa)
     => ( 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),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ) ) ).

% self_le_power
tff(fact_3679_one__less__power,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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,Aa),Nb)) ) ) ).

% one_less_power
tff(fact_3680_one__less__power,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => ( 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(Aa),Nb)) ) ) ).

% one_less_power
tff(fact_3681_one__less__power,axiom,
    ! [Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => ( 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,Aa),Nb)) ) ) ).

% one_less_power
tff(fact_3682_one__less__power,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => ( 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,Aa),Nb)) ) ) ).

% one_less_power
tff(fact_3683_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_3684_power__diff,axiom,
    ! [Aa: complex,Nb: nat,Mb: nat] :
      ( ( Aa != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)) ) ) ) ).

% power_diff
tff(fact_3685_power__diff,axiom,
    ! [Aa: real,Nb: nat,Mb: nat] :
      ( ( Aa != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ) ) ) ).

% power_diff
tff(fact_3686_power__diff,axiom,
    ! [Aa: rat,Nb: nat,Mb: nat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(power_power_rat(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(Aa),Mb)),aa_nat_rat(power_power_rat(Aa),Nb)) ) ) ) ).

% power_diff
tff(fact_3687_power__diff,axiom,
    ! [Aa: nat,Nb: nat,Mb: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) ) ) ) ).

% power_diff
tff(fact_3688_power__diff,axiom,
    ! [Aa: int,Nb: nat,Mb: nat] :
      ( ( Aa != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ) ) ) ).

% power_diff
tff(fact_3689_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_3690_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3691_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_3692_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_3693_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_3694_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_3695_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_3696_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_3697_split__div,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I5)),J3) )
               => aa_nat_o(P,I5) ) ) ) ) ) ).

% split_div
tff(fact_3698_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Mb: 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,Mb),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,Mb),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3699_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y3) )
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => ( (Y3)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A2),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Y3) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)
               => ( (Y3)
                <=> ~ $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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3700_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A2),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)
             => ~ $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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3701_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A2),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)
               => $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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3702_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_3703_vebt__delete_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),zero_zero_nat,TrLst,Smry),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),zero_zero_nat,TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_3704_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_3705_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_3706_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_3707_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve2) = none_nat ).

% vebt_succ.simps(4)
tff(fact_3708_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve2),Vf2) = none_nat ).

% vebt_pred.simps(5)
tff(fact_3709_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,Ve2: vEBT_VEBT,Vf2: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve2)),Vf2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_3710_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,Ve2: vEBT_VEBT,Vf2: nat] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve2)),Vf2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_3711_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd)),Ve2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_3712_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd)),Ve2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_3713_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y3) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Y3) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Y3) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( (Y3)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( (Y3)
                  <=> ~ ( ( 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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
                   => ( (Y3)
                    <=> ~ $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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3714_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = 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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3715_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,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xb) = Y3 )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ( Y3 != one_one_nat ) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => ( Y3 != one_one_nat ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ( Y3 != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => ( Y3 != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_3716_convex__bound__lt,axiom,
    ! [Xb: real,Aa: real,Y3: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Aa)
       => ( 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(plus_plus_real(U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),Xb)),aa_real_real(times_times_real(V),Y3))),Aa) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3717_convex__bound__lt,axiom,
    ! [Xb: rat,Aa: rat,Y3: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Aa)
       => ( 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),Y3))),Aa) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3718_convex__bound__lt,axiom,
    ! [Xb: int,Aa: int,Y3: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Aa)
       => ( 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),Y3))),Aa) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3719_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,aa_num_real(numeral_numeral_real,W)),aa_real_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(times_times_real(aa_num_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(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3720_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,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_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(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3721_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(divide_divide_real(B),C)),aa_num_real(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(times_times_real(aa_num_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(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3722_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_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(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3723_half__gt__zero__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Aa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ).

% half_gt_zero_iff
tff(fact_3724_half__gt__zero__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ).

% half_gt_zero_iff
tff(fact_3725_half__gt__zero,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Aa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_3726_half__gt__zero,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_3727_power2__le__imp__le,axiom,
    ! [Xb: real,Y3: 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,Y3),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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ).

% power2_le_imp_le
tff(fact_3728_power2__le__imp__le,axiom,
    ! [Xb: rat,Y3: 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(Y3),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),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3) ) ) ).

% power2_le_imp_le
tff(fact_3729_power2__le__imp__le,axiom,
    ! [Xb: nat,Y3: 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,Y3),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),Y3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3) ) ) ).

% power2_le_imp_le
tff(fact_3730_power2__le__imp__le,axiom,
    ! [Xb: int,Y3: 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,Y3),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),Y3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3) ) ) ).

% power2_le_imp_le
tff(fact_3731_power2__eq__imp__eq,axiom,
    ! [Xb: real,Y3: 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,Y3),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),Y3)
         => ( Xb = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3732_power2__eq__imp__eq,axiom,
    ! [Xb: rat,Y3: 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(Y3),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),Y3)
         => ( Xb = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3733_power2__eq__imp__eq,axiom,
    ! [Xb: nat,Y3: 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,Y3),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),Y3)
         => ( Xb = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3734_power2__eq__imp__eq,axiom,
    ! [Xb: int,Y3: 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,Y3),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),Y3)
         => ( Xb = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3735_zero__le__power2,axiom,
    ! [Aa: 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3736_zero__le__power2,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3737_zero__le__power2,axiom,
    ! [Aa: 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3738_power2__less__0,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_3739_power2__less__0,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_3740_power2__less__0,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_3741_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3742_exp__add__not__zero__imp__right,axiom,
    ! [Mb: 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,Mb),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_3743_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3744_exp__add__not__zero__imp__left,axiom,
    ! [Mb: 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,Mb),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))),Mb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3745_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3746_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: 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),Mb)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3747_power__diff__power__eq,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] :
      ( ( Aa != 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,Aa),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_3748_power__diff__power__eq,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] :
      ( ( Aa != 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,Aa),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_3749_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_3750_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_3751_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_3752_power__eq__if,axiom,
    ! [P2: complex,Mb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,P2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3753_power__eq__if,axiom,
    ! [P2: real,Mb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_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,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3754_power__eq__if,axiom,
    ! [P2: rat,Mb: nat] :
      aa_nat_rat(power_power_rat(P2),Mb) = $ite(Mb = 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,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3755_power__eq__if,axiom,
    ! [P2: nat,Mb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),Mb) = $ite(Mb = 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,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3756_power__eq__if,axiom,
    ! [P2: int,Mb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),Mb) = $ite(Mb = 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,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3757_power__minus__mult,axiom,
    ! [Nb: nat,Aa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),Aa) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb) ) ) ).

% power_minus_mult
tff(fact_3758_power__minus__mult,axiom,
    ! [Nb: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),Aa) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb) ) ) ).

% power_minus_mult
tff(fact_3759_power__minus__mult,axiom,
    ! [Nb: nat,Aa: 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(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),Aa) = aa_nat_rat(power_power_rat(Aa),Nb) ) ) ).

% power_minus_mult
tff(fact_3760_power__minus__mult,axiom,
    ! [Nb: nat,Aa: 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb) ) ) ).

% power_minus_mult
tff(fact_3761_power__minus__mult,axiom,
    ! [Nb: nat,Aa: 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),Aa) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb) ) ) ).

% power_minus_mult
tff(fact_3762_split__div_H,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),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)),Mb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_3763_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_3764_vebt__delete_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_3765_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(vEBT_T_d_e_l_e_t_e(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_3766_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2) = none_nat ).

% vebt_succ.simps(5)
tff(fact_3767_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2) = none_nat ).

% vebt_pred.simps(6)
tff(fact_3768_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,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)),Vj2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_3769_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,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)),Vj2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_3770_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,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)),Vi2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_3771_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,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)),Vi2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_3772_power2__less__imp__less,axiom,
    ! [Xb: real,Y3: 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,Y3),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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ) ).

% power2_less_imp_less
tff(fact_3773_power2__less__imp__less,axiom,
    ! [Xb: rat,Y3: 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(Y3),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),Y3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3) ) ) ).

% power2_less_imp_less
tff(fact_3774_power2__less__imp__less,axiom,
    ! [Xb: nat,Y3: 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,Y3),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),Y3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3) ) ) ).

% power2_less_imp_less
tff(fact_3775_power2__less__imp__less,axiom,
    ! [Xb: int,Y3: 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,Y3),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),Y3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3) ) ) ).

% power2_less_imp_less
tff(fact_3776_sum__power2__le__zero__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3777_sum__power2__le__zero__iff,axiom,
    ! [Xb: rat,Y3: 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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3778_sum__power2__le__zero__iff,axiom,
    ! [Xb: int,Y3: 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3779_sum__power2__ge__zero,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3780_sum__power2__ge__zero,axiom,
    ! [Xb: rat,Y3: 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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3781_sum__power2__ge__zero,axiom,
    ! [Xb: int,Y3: 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3782_sum__power2__gt__zero__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3783_sum__power2__gt__zero__iff,axiom,
    ! [Xb: rat,Y3: 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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3784_sum__power2__gt__zero__iff,axiom,
    ! [Xb: int,Y3: 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3785_not__sum__power2__lt__zero,axiom,
    ! [Xb: real,Y3: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_3786_not__sum__power2__lt__zero,axiom,
    ! [Xb: rat,Y3: 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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_3787_not__sum__power2__lt__zero,axiom,
    ! [Xb: int,Y3: 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_3788_zero__le__even__power_H,axiom,
    ! [Aa: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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_3789_zero__le__even__power_H,axiom,
    ! [Aa: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),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_3790_zero__le__even__power_H,axiom,
    ! [Aa: 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,Aa),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_3791_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_3792_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_3793_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => 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))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => 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))),N))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_3794_divide__divide__eq__left_H,axiom,
    ! [Aa: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B)) ).

% divide_divide_eq_left'
tff(fact_3795_divide__divide__eq__left_H,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(Aa),B)),C) = aa_real_real(divide_divide_real(Aa),aa_real_real(times_times_real(C),B)) ).

% divide_divide_eq_left'
tff(fact_3796_divide__divide__eq__left_H,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(Aa),B)),C) = aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_3797_divide__divide__times__eq,axiom,
    ! [Xb: complex,Y3: complex,Z: complex,W: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),W)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),W)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z)) ).

% divide_divide_times_eq
tff(fact_3798_divide__divide__times__eq,axiom,
    ! [Xb: real,Y3: real,Z: real,W: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(Xb),Y3)),aa_real_real(divide_divide_real(Z),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xb),W)),aa_real_real(times_times_real(Y3),Z)) ).

% divide_divide_times_eq
tff(fact_3799_divide__divide__times__eq,axiom,
    ! [Xb: rat,Y3: rat,Z: rat,W: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(Xb),Y3)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xb),W)),aa_rat_rat(times_times_rat(Y3),Z)) ).

% divide_divide_times_eq
tff(fact_3800_times__divide__times__eq,axiom,
    ! [Xb: complex,Y3: complex,Z: complex,W: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),W)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),W)) ).

% times_divide_times_eq
tff(fact_3801_times__divide__times__eq,axiom,
    ! [Xb: real,Y3: real,Z: real,W: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(Xb),Y3)),aa_real_real(divide_divide_real(Z),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(Y3),W)) ).

% times_divide_times_eq
tff(fact_3802_times__divide__times__eq,axiom,
    ! [Xb: rat,Y3: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(Xb),Y3)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y3),W)) ).

% times_divide_times_eq
tff(fact_3803_add__divide__distrib,axiom,
    ! [Aa: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),C)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) ).

% add_divide_distrib
tff(fact_3804_add__divide__distrib,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Aa),B)),C) = aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Aa),C)),aa_real_real(divide_divide_real(B),C)) ).

% add_divide_distrib
tff(fact_3805_add__divide__distrib,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Aa),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Aa),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% add_divide_distrib
tff(fact_3806_diff__divide__distrib,axiom,
    ! [Aa: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),C)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) ).

% diff_divide_distrib
tff(fact_3807_diff__divide__distrib,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Aa),B)),C) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Aa),C)),aa_real_real(divide_divide_real(B),C)) ).

% diff_divide_distrib
tff(fact_3808_diff__divide__distrib,axiom,
    ! [Aa: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Aa),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Aa),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% diff_divide_distrib
tff(fact_3809_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(vEBT_T_d_e_l_e_t_e(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_3810_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(vEBT_T_m_e_m_b_e_r(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_3811_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xb) = Y3 )
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((B3),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 = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y3 != one_one_nat ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2,Uz2)
             => ( Y3 != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_3812_odd__0__le__power__imp__0__le,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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),Aa) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3813_odd__0__le__power__imp__0__le,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),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),Aa) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3814_odd__0__le__power__imp__0__le,axiom,
    ! [Aa: 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,Aa),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),Aa) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3815_odd__power__less__zero,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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,Aa),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_3816_odd__power__less__zero,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),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_3817_odd__power__less__zero,axiom,
    ! [Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),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_3818_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_3819_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_3820_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A2),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3821_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),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_3822_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xb = 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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3823_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y3) )
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => ( (Y3)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A2),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Y3) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => (Y3) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Y3) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = 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)
                   => ( (Y3)
                    <=> ~ $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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3824_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A2),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3825_arith__geo__mean,axiom,
    ! [U: real,Xb: real,Y3: 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(times_times_real(Xb),Y3) )
     => ( 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),Y3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Y3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3826_arith__geo__mean,axiom,
    ! [U: rat,Xb: rat,Y3: 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),Y3) )
     => ( 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),Y3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Y3)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3827_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xb),Xaa) = Y3 )
     => ( ( ? [A2: $o,B3: $o] : Xb = vEBT_Leaf((A2),(B3))
         => ( Y3 != 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 = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => ( Y3 != 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 = 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)
                   => ( Y3 != 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(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_3828_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xb),Xaa) = Y3 )
     => ( ( ? [A2: $o,B3: $o] : Xb = vEBT_Leaf((A2),(B3))
         => ( Y3 != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] : Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S2)
           => ( Y3 != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] : Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary)
               => ( Y3 != one_one_nat ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = 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) )
                   => ( Y3 != $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(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(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_3829_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A3: $o,B4: $o] : A1 = vEBT_Leaf((A3),(B4))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = 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(vEBT_invar_vebt(X),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( 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))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = 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(vEBT_invar_vebt(X),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( 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,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = 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(vEBT_invar_vebt(X),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( 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))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
               => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I5) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I5: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N4) = I5 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N4) = I5 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N4)) )
                       => ( 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,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = 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(vEBT_invar_vebt(X),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( 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,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_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,N4)))
               => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I5) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I5: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_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,N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N4) = I5 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N4) = I5 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N4)) )
                       => ( 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_3830_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A2: $o,B3: $o] : A1 = vEBT_Leaf((A2),(B3))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X3),N) )
                 => ( aa_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 = N )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                         => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X3),N) )
                   => ( aa_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,N) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                           => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X3),N) )
                     => ( aa_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 = N )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),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_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_13)
                                    <=> aa_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_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I4: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                & ! [X3: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I4 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)) )
                                                   => ( 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,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X3: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X3),N) )
                       => ( aa_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,N) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),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_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_13)
                                      <=> aa_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_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I4: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I4 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)) )
                                                     => ( 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_3831_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y3 )
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => ( Y3 != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B3))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S2) )
             => ( Y3 != vEBT_Node(Info2,zero_zero_nat,Ts,S2) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2) )
               => ( Y3 != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                 => ( Y3 != 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 = 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) )
                   => ( Y3 != $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 ) ) ),
                              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)),
                              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_3832_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xb),Xaa) = Y3 )
     => ( ( ? [A2: $o,B3: $o] : Xb = vEBT_Leaf((A2),(B3))
         => ( Y3 != 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,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] : Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S2)
           => ( Y3 != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] : Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary)
               => ( Y3 != 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 = 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) )
                   => ( Y3 != 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(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(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_3833_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xb),Xaa) = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y3 != one_one_nat ) ) )
       => ( ( ? [A2: $o,Uw2: $o] : Xb = vEBT_Leaf((A2),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y3 != one_one_nat ) ) )
         => ( ( ? [A2: $o,B3: $o] : Xb = vEBT_Leaf((A2),(B3))
             => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
               => ( Y3 != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve)
                 => ( Y3 != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)
                   => ( Y3 != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = 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) )
                       => ( Y3 != $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,some_nat(l)),minlow) ),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_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(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_3834_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xb),Xaa) = Y3 )
     => ( ( ? [Uu2: $o,B3: $o] : Xb = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero_nat )
           => ( Y3 != one_one_nat ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y3 != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
                 => ( Y3 != one_one_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = 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) )
                     => ( Y3 != $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,some_nat(l)),maxlow) ),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_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(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_3835_vebt__delete_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y3 )
     => ( ! [A2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B3)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y3 != vEBT_Leaf($false,(B3)) ) ) )
       => ( ! [A2: $o] :
              ( ? [B3: $o] : Xb = vEBT_Leaf((A2),(B3))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y3 != vEBT_Leaf((A2),$false) ) ) )
         => ( ! [A2: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B3)) )
               => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
                 => ( Y3 != vEBT_Leaf((A2),(B3)) ) ) )
           => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
                 => ( Y3 != vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) ) )
             => ( ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst2,Smry2) )
                   => ( Y3 != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst2,Smry2) ) )
               => ( ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                     => ( Y3 != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = 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) )
                       => ( Y3 != $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) ),
                              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 ) ),
                                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),
                                                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) ),
                                              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) ) ) ),
                                        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_3836_less__half__sum,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Aa),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_3837_less__half__sum,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Aa),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_3838_gt__half__sum,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Aa),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_3839_gt__half__sum,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Aa),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_3840_vebt__pred_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: option_nat] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y3 != none_nat ) ) )
       => ( ! [A2: $o] :
              ( ? [Uw2: $o] : Xb = vEBT_Leaf((A2),(Uw2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y3 != $ite((A2),some_nat(zero_zero_nat),none_nat) ) ) )
         => ( ! [A2: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B3)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y3 != $ite(
                        (B3),
                        some_nat(one_one_nat),
                        $ite((A2),some_nat(zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y3 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve)
                 => ( Y3 != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)
                   => ( Y3 != none_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = 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) )
                       => ( Y3 != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              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,some_nat(l)),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= vEBT_vebt_pred(Summary,h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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_3841_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: option_nat] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y3 )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y3 != $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y3 != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y3 != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
               => ( Y3 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
                 => ( Y3 != none_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = 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) )
                     => ( Y3 != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            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,some_nat(l)),maxlow) ),
                                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= 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,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_3842_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(Xb),Xaa) = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y3 != one_one_nat ) ) )
       => ( ( ? [A2: $o,Uw2: $o] : Xb = vEBT_Leaf((A2),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
         => ( ! [A2: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B3)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $ite((B3),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve)
                 => ( Y3 != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)
                   => ( Y3 != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = 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) )
                       => ( Y3 != 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,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(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList,h)),l)),
                                              $let(
                                                pr: option_nat,
                                                pr:= 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(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_3843_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(Xb),Xaa) = Y3 )
     => ( ( ? [Uu2: $o,B3: $o] : Xb = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero_nat )
           => ( Y3 != 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))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y3 != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
                 => ( Y3 != one_one_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = 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) )
                     => ( Y3 != 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,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(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList,h)),l)),
                                              $let(
                                                sc: option_nat,
                                                sc:= 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(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_3844_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xb),Xaa) = Y3 )
     => ( ( ? [A2: $o,B3: $o] : Xb = vEBT_Leaf((A2),(B3))
         => ( Y3 != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y3 != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => ( Y3 != one_one_nat ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = 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)
                   => ( Y3 != 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(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_3845_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,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r2(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,Va)),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,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(TreeLista)),aa_nat_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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                  zero_zero_nat ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_3846_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_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_3847_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(Xb),Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y3 = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),zero_zero_nat)) ) ) )
         => ( ! [A2: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B3)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A2: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y3 = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList,Summary) )
                     => ( ( Y3 = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),produc738532404422230701BT_nat(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 = 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) )
                       => ( ( Y3 = one_one_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),produc738532404422230701BT_nat(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 = 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) )
                         => ( ( Y3 = 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(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(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),produc738532404422230701BT_nat(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_3848_set__bit__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),Aa) = 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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3849_set__bit__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),Aa) = 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3850_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(Xb),Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y3 = 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),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
                     => ( ( Y3 = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = 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) )
                       => ( ( Y3 = 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,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(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList,h)),l)),
                                                  $let(
                                                    sc: option_nat,
                                                    sc:= 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(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),produc738532404422230701BT_nat(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_3851_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(Xb),Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y3 = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = 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),produc738532404422230701BT_nat(vEBT_Leaf((A2),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A2: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B3)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite((B3),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),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve) )
                     => ( ( Y3 = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi) )
                       => ( ( Y3 = one_one_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = 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) )
                         => ( ( Y3 = 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,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(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList,h)),l)),
                                                  $let(
                                                    pr: option_nat,
                                                    pr:= 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(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),produc738532404422230701BT_nat(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_3852_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: option_nat] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y3 = $ite((B3),some_nat(one_one_nat),none_nat) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y3 = none_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y3 = none_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y3 = none_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
                     => ( ( Y3 = none_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = 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) )
                       => ( ( Y3 = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                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,some_nat(l)),maxlow) ),
                                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= 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,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),produc738532404422230701BT_nat(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_3853_vebt__pred_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: option_nat] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y3 = none_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = $ite((A2),some_nat(zero_zero_nat),none_nat) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A2: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B3)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y3 = $ite(
                              (B3),
                              some_nat(one_one_nat),
                              $ite((A2),some_nat(zero_zero_nat),none_nat) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y3 = none_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve) )
                     => ( ( Y3 = none_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi) )
                       => ( ( Y3 = none_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = 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) )
                         => ( ( Y3 = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  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,some_nat(l)),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= vEBT_vebt_pred(Summary,h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,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),produc738532404422230701BT_nat(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_3854_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_3855_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_3856_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(times_times_real(Xb),Xb))
    <=> ( Xb = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_3857_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_3858_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_3859_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_3860_zdiv__zmult2__eq,axiom,
    ! [C: int,Aa: 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,Aa),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,Aa),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_3861_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3862_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,Aa: 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,Aa),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3863_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,Aa: 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,Aa),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3864_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_3865_div__nonpos__pos__le0,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,Aa),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_3866_div__nonneg__neg__le0,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_3867_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_3868_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_3869_zdiv__mono2__neg,axiom,
    ! [Aa: int,B5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B5)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3870_zdiv__mono1__neg,axiom,
    ! [Aa: int,A4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),A4)
     => ( 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,A4),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_3871_int__div__pos__eq,axiom,
    ! [Aa: int,B: int,Q2: int,R2: int] :
      ( ( Aa = 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,Aa),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3872_int__div__neg__eq,axiom,
    ! [Aa: int,B: int,Q2: int,R2: int] :
      ( ( Aa = 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,Aa),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3873_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_3874_zdiv__mono2,axiom,
    ! [Aa: int,B5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B5)) ) ) ) ).

% zdiv_mono2
tff(fact_3875_zdiv__mono1,axiom,
    ! [Aa: int,A4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),A4)
     => ( 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,Aa),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B)) ) ) ).

% zdiv_mono1
tff(fact_3876_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)
         => ! [I5: 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),I5)),J3) ) )
             => aa_int_o(P,I5) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I5: 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),I5)),J3) ) )
             => aa_int_o(P,I5) ) ) ) ) ).

% split_zdiv
tff(fact_3877_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_3878_enat__0__less__mult__iff,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,times_7803423173614009249d_enat(Mb,Nb))
    <=> ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Mb)
        & ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_3879_i0__lb,axiom,
    ! [Nb: extended_enat] : ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat,Nb) ).

% i0_lb
tff(fact_3880_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Nb,zero_z5237406670263579293d_enat)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_3881_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [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_3882_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [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_3883_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_3884_realpow__pos__nth2,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,Nb)) = Aa ) ) ) ).

% realpow_pos_nth2
tff(fact_3885_real__arch__pow__inv,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)),Y3) ) ) ).

% real_arch_pow_inv
tff(fact_3886_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,
    ! [Aa: $o,B: $o,Xb: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((Aa),(B))),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_3887_realpow__pos__nth,axiom,
    ! [Nb: nat,Aa: 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)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),Nb) = Aa ) ) ) ) ).

% realpow_pos_nth
tff(fact_3888_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,Aa: 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)
       => ? [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) = Aa )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Nb) = Aa ) )
               => ( Y2 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3889_neg__zdiv__mult__2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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))),Aa)) = 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)),Aa) ) ) ).

% neg_zdiv_mult_2
tff(fact_3890_pos__zdiv__mult__2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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))),Aa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Aa) ) ) ).

% pos_zdiv_mult_2
tff(fact_3891_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(vEBT_T_m_e_m_b_e_r2(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_3892_int__power__div__base,axiom,
    ! [Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,K),Mb)),K) = aa_nat_int(aa_int_fun_nat_int(power_power_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_3893_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(vEBT_T_m_e_m_b_e_r2(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_3894_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_3895_member__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_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_3896_vebt__delete_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y3 = vEBT_Leaf($false,(B3)) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),zero_zero_nat)) ) ) )
         => ( ! [A2: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B3)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = vEBT_Leaf((A2),$false) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A2: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y3 = vEBT_Leaf((A2),(B3)) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
                   => ( ( Y3 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst2,Smry2) )
                     => ( ( Y3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst2,Smry2) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst2,Smry2),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                       => ( ( Y3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = 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) )
                         => ( ( Y3 = $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) ),
                                  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 ) ),
                                    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),
                                                    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) ),
                                                  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) ) ) ),
                                            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),produc738532404422230701BT_nat(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_3897_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xb),Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y3 = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
                     => ( ( Y3 = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = 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) )
                       => ( ( Y3 = $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,some_nat(l)),maxlow) ),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_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(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),produc738532404422230701BT_nat(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_3898_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xb),Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = 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),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S2) )
               => ( ( Y3 = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S2),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                   => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( Y3 = 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(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(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),produc738532404422230701BT_nat(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_3899_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xb),Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y3 = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A2: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B3)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y3 = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve) )
                     => ( ( Y3 = one_one_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi) )
                       => ( ( Y3 = one_one_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = 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) )
                         => ( ( Y3 = $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,some_nat(l)),minlow) ),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_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(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),produc738532404422230701BT_nat(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_3900_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B3))) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S2) )
               => ( ( Y3 = vEBT_Node(Info2,zero_zero_nat,Ts,S2) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S2),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2) )
                 => ( ( Y3 = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                   => ( ( Y3 = 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),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( Y3 = $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 ) ) ),
                                  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)),
                                  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),produc738532404422230701BT_nat(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_3901_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xb),Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = one_one_nat )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S2) )
               => ( ( Y3 = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S2),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S2),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( Y3 = $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(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(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),produc738532404422230701BT_nat(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_3902_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xb),Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = 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),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( Y3 = 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(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),produc738532404422230701BT_nat(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_3903_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,Y3: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xb),Xaa) = Y3 )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = one_one_nat )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y3 = one_one_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( Y3 = 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(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),produc738532404422230701BT_nat(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_3904_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q2),zero_z5237406670263579293d_enat) = Q2 ).

% max_enat_simps(2)
tff(fact_3905_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_3906_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_3907_set__bit__negative__int__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_se7879613467334960850it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_3908_q__pos__lemma,axiom,
    ! [B5: int,Q5: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_3909_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4) )
     => ( 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,B5),Q5)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B5)
         => ( 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),B5)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3910_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4) )
     => ( 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,B5),Q5)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3911_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: 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)),R4)),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),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3912_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: 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)),R4)),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),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3913_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_3914_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_3915_imult__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( times_7803423173614009249d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_3916_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_3917_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_3918_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y3) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( (Y3)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A2),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Y3)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Y3)
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Y3)
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( (Y3)
                        <=> $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)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(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_3919_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A2),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(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),produc738532404422230701BT_nat(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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3920_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A2),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3921_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A2),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3922_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y3) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( (Y3)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A2),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Y3)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
                 => ( ( (Y3)
                    <=> $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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3923_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A2),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(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),produc738532404422230701BT_nat(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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3924_double__eq__0__iff,axiom,
    ! [Aa: real] :
      ( ( aa_real_real(plus_plus_real(Aa),Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_3925_double__eq__0__iff,axiom,
    ! [Aa: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_3926_double__eq__0__iff,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Aa) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_3927_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_3928_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_3929_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_3930_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_3931_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_3932_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_3933_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_3934_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_3935_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_3936_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_3937_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_3938_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_3939_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_3940_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_3941_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_3942_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
      <=> ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3943_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_3944_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_3945_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_3946_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(P,I2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_3947_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_3948_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3949_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y3) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y3)
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ (Y3)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( ( (Y3)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( ( (Y3)
                      <=> ( ( 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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                     => ( ( (Y3)
                        <=> $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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3950_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3951_atLeastatMost__empty,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Aa)
     => ( set_or633870826150836451st_rat(Aa,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_3952_atLeastatMost__empty,axiom,
    ! [B: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),Aa)
     => ( set_or7049704709247886629st_num(Aa,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_3953_atLeastatMost__empty,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa)
     => ( set_or1269000886237332187st_nat(Aa,B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_3954_atLeastatMost__empty,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Aa)
     => ( set_or1266510415728281911st_int(Aa,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_3955_atLeastatMost__empty,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => ( set_or1222579329274155063t_real(Aa,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_3956_atLeastatMost__subset__iff,axiom,
    ! [Aa: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(Aa,B),set_or370866239135849197et_int(C,D))
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(Aa),B)
        | ( aa_set_int_o(ord_less_eq_set_int(C),Aa)
          & aa_set_int_o(ord_less_eq_set_int(B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3957_atLeastatMost__subset__iff,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(Aa,B)),set_or633870826150836451st_rat(C,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3958_atLeastatMost__subset__iff,axiom,
    ! [Aa: num,B: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(Aa,B)),set_or7049704709247886629st_num(C,D))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3959_atLeastatMost__subset__iff,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_or1269000886237332187st_nat(Aa,B)),set_or1269000886237332187st_nat(C,D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3960_atLeastatMost__subset__iff,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(Aa,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3961_atLeastatMost__subset__iff,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(Aa,B)),set_or1222579329274155063t_real(C,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3962_atLeastatMost__empty__iff,axiom,
    ! [Aa: set_int,B: set_int] :
      ( ( set_or370866239135849197et_int(Aa,B) = bot_bot_set_set_int )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(Aa),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3963_atLeastatMost__empty__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(Aa,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3964_atLeastatMost__empty__iff,axiom,
    ! [Aa: num,B: num] :
      ( ( set_or7049704709247886629st_num(Aa,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3965_atLeastatMost__empty__iff,axiom,
    ! [Aa: nat,B: nat] :
      ( ( set_or1269000886237332187st_nat(Aa,B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3966_atLeastatMost__empty__iff,axiom,
    ! [Aa: int,B: int] :
      ( ( set_or1266510415728281911st_int(Aa,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3967_atLeastatMost__empty__iff,axiom,
    ! [Aa: real,B: real] :
      ( ( set_or1222579329274155063t_real(Aa,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3968_atLeastatMost__empty__iff2,axiom,
    ! [Aa: set_int,B: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(Aa,B) )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(Aa),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3969_atLeastatMost__empty__iff2,axiom,
    ! [Aa: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(Aa,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3970_atLeastatMost__empty__iff2,axiom,
    ! [Aa: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(Aa,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3971_atLeastatMost__empty__iff2,axiom,
    ! [Aa: nat,B: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(Aa,B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3972_atLeastatMost__empty__iff2,axiom,
    ! [Aa: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(Aa,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3973_atLeastatMost__empty__iff2,axiom,
    ! [Aa: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(Aa,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3974_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_3975_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I),set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I)
        & aa_set_nat_o(ord_less_eq_set_nat(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3976_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_3977_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_3978_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( aa_set_num_o(member_num(I),set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3979_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_3980_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_3981_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_3982_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_int_o(ord_less_eq_set_int(L),H)
          & ~ aa_set_int_o(ord_less_eq_set_int(L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3983_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3984_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
          & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3985_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3986_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3987_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
          & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3988_atLeastAtMostPlus1__int__conv,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb))
     => ( set_or1266510415728281911st_int(Mb,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb),set_or1266510415728281911st_int(Mb,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_3989_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_3990_bset_I1_J,axiom,
    ! [D2: int,B2: 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,D2))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( 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),D2)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( 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),D2)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( 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),D2))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D2)) ) ) ) ) ) ).

% bset(1)
tff(fact_3991_bset_I2_J,axiom,
    ! [D2: int,B2: 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,D2))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( 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),D2)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( 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),D2)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( 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),D2))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D2)) ) ) ) ) ) ).

% bset(2)
tff(fact_3992_aset_I1_J,axiom,
    ! [D2: int,A: 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,D2))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A)
                 => ( 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),D2)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A)
                   => ( 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),D2)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A)
                   => ( 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),D2))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D2)) ) ) ) ) ) ).

% aset(1)
tff(fact_3993_aset_I2_J,axiom,
    ! [D2: int,A: 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,D2))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A)
                 => ( 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),D2)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A)
                   => ( 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),D2)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A)
                   => ( 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),D2))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D2)) ) ) ) ) ) ).

% aset(2)
tff(fact_3994_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_3995_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_3996_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X3) ) ).

% minf(7)
tff(fact_3997_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_3998_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_3999_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_4000_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_4001_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Ta) ) ).

% minf(5)
tff(fact_4002_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_4003_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_4004_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_4005_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_4006_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_4007_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_4008_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_4009_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_4010_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_4011_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_4012_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_4013_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_4014_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_4015_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_4016_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(aa_num_fun_num_o(ord_less_num,X2),Z4)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_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_4017_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_4018_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_4019_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_4020_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_4021_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(aa_num_fun_num_o(ord_less_num,X2),Z4)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_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_4022_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_4023_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_4024_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_4025_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_4026_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X3) ) ).

% pinf(7)
tff(fact_4027_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_4028_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_4029_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_4030_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_4031_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Ta) ) ).

% pinf(5)
tff(fact_4032_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_4033_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_4034_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_4035_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_4036_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_4037_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_4038_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_4039_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_4040_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_4041_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_4042_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_4043_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_4044_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_4045_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_4046_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(aa_num_fun_num_o(ord_less_num,Z4),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_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_4047_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_4048_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_4049_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_4050_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_4051_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(aa_num_fun_num_o(ord_less_num,Z4),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_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_4052_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_4053_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_4054_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xb: nat,M6: 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),M6) )
       => ~ ! [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_4055_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc8114805097830739791at_num] :
      ~ ! [F2: fun_nat_fun_num_num,A2: nat,B3: nat,Acc: num] : Xb != produc460779904311030791at_num(F2,produc1195630363706982562at_num(A2,product_Pair_nat_num(B3,Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_4056_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A2: nat,B3: nat,Acc: nat] : Xb != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A2,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_4057_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_13: int] : aa_int_o(P,X_13)
        <=> ? [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_4058_aset_I7_J,axiom,
    ! [D2: int,A: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A)
                 => ( 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),D2)) ) ) ) ).

% aset(7)
tff(fact_4059_aset_I5_J,axiom,
    ! [D2: int,Ta: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(Ta),A)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A)
                   => ( 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),D2)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_4060_aset_I4_J,axiom,
    ! [D2: int,Ta: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(Ta),A)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A)
                   => ( 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),D2) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_4061_aset_I3_J,axiom,
    ! [D2: int,Ta: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A)
                   => ( 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),D2) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_4062_bset_I7_J,axiom,
    ! [D2: int,Ta: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(Ta),B2)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( 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),D2)) ) ) ) ) ).

% bset(7)
tff(fact_4063_bset_I5_J,axiom,
    ! [D2: int,B2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B2)
                 => ( 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),D2)),Ta) ) ) ) ).

% bset(5)
tff(fact_4064_bset_I4_J,axiom,
    ! [D2: int,Ta: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(Ta),B2)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( 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),D2) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_4065_bset_I3_J,axiom,
    ! [D2: int,Ta: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B2)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( 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),D2) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_4066_aset_I8_J,axiom,
    ! [D2: int,A: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A)
                 => ( 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),D2)) ) ) ) ).

% aset(8)
tff(fact_4067_aset_I6_J,axiom,
    ! [D2: int,Ta: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A)
                   => ( 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),D2)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_4068_bset_I8_J,axiom,
    ! [D2: int,Ta: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B2)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B2)
                   => ( 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),D2)) ) ) ) ) ).

% bset(8)
tff(fact_4069_bset_I6_J,axiom,
    ! [D2: int,B2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B2)
                 => ( 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),D2)),Ta) ) ) ) ).

% bset(6)
tff(fact_4070_cppi,axiom,
    ! [D2: int,P: fun_int_o,P3: fun_int_o,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ? [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,D2))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),A)
                     => ( 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),D2)) ) )
         => ( ! [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),D2))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D2))
                    & aa_int_o(P3,X) )
                | ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D2))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4071_cpmi,axiom,
    ! [D2: int,P: fun_int_o,P3: fun_int_o,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ? [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,D2))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),B2)
                     => ( 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),D2)) ) )
         => ( ! [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),D2))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D2))
                    & aa_int_o(P3,X) )
                | ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D2))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4072_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_4073_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_4074_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Ta) ) ).

% pinf(6)
tff(fact_4075_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_4076_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_4077_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_4078_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_4079_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X3) ) ).

% pinf(8)
tff(fact_4080_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_4081_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_4082_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_4083_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_4084_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Ta) ) ).

% minf(6)
tff(fact_4085_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_4086_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_4087_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_4088_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_4089_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X3) ) ).

% minf(8)
tff(fact_4090_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_4091_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_4092_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D2: real,Q: fun_real_o] :
      ( ! [X2: real,K3: real] :
          ( aa_real_o(P,X2)
        <=> aa_real_o(P,aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K3),D2))) )
     => ( ! [X2: real,K3: real] :
            ( aa_real_o(Q,X2)
          <=> aa_real_o(Q,aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K3),D2))) )
       => ! [X3: real,K4: real] :
            ( ( aa_real_o(P,X3)
              | aa_real_o(Q,X3) )
          <=> ( aa_real_o(P,aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K4),D2)))
              | aa_real_o(Q,aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K4),D2))) ) ) ) ) ).

% inf_period(2)
tff(fact_4093_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D2: rat,Q: fun_rat_o] :
      ( ! [X2: rat,K3: rat] :
          ( aa_rat_o(P,X2)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K3),D2))) )
     => ( ! [X2: rat,K3: rat] :
            ( aa_rat_o(Q,X2)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K3),D2))) )
       => ! [X3: rat,K4: rat] :
            ( ( aa_rat_o(P,X3)
              | aa_rat_o(Q,X3) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D2)))
              | aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D2))) ) ) ) ) ).

% inf_period(2)
tff(fact_4094_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D2: 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),D2))) )
     => ( ! [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),D2))) )
       => ! [X3: int,K4: 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,K4),D2)))
              | 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,K4),D2))) ) ) ) ) ).

% inf_period(2)
tff(fact_4095_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D2: real,Q: fun_real_o] :
      ( ! [X2: real,K3: real] :
          ( aa_real_o(P,X2)
        <=> aa_real_o(P,aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K3),D2))) )
     => ( ! [X2: real,K3: real] :
            ( aa_real_o(Q,X2)
          <=> aa_real_o(Q,aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K3),D2))) )
       => ! [X3: real,K4: real] :
            ( ( aa_real_o(P,X3)
              & aa_real_o(Q,X3) )
          <=> ( aa_real_o(P,aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K4),D2)))
              & aa_real_o(Q,aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K4),D2))) ) ) ) ) ).

% inf_period(1)
tff(fact_4096_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D2: rat,Q: fun_rat_o] :
      ( ! [X2: rat,K3: rat] :
          ( aa_rat_o(P,X2)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K3),D2))) )
     => ( ! [X2: rat,K3: rat] :
            ( aa_rat_o(Q,X2)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K3),D2))) )
       => ! [X3: rat,K4: rat] :
            ( ( aa_rat_o(P,X3)
              & aa_rat_o(Q,X3) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D2)))
              & aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D2))) ) ) ) ) ).

% inf_period(1)
tff(fact_4097_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D2: 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),D2))) )
     => ( ! [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),D2))) )
       => ! [X3: int,K4: 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,K4),D2)))
              & 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,K4),D2))) ) ) ) ) ).

% inf_period(1)
tff(fact_4098_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_4099_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_4100_atLeastatMost__psubset__iff,axiom,
    ! [Aa: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(Aa,B),set_or370866239135849197et_int(C,D))
    <=> ( ( ~ aa_set_int_o(ord_less_eq_set_int(Aa),B)
          | ( aa_set_int_o(ord_less_eq_set_int(C),Aa)
            & aa_set_int_o(ord_less_eq_set_int(B),D)
            & ( aa_set_int_o(ord_less_set_int(C),Aa)
              | 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_4101_atLeastatMost__psubset__iff,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(Aa,B)),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
            & 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),Aa)
              | 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_4102_atLeastatMost__psubset__iff,axiom,
    ! [Aa: num,B: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(Aa,B)),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4103_atLeastatMost__psubset__iff,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(set_or1269000886237332187st_nat(Aa,B)),set_or1269000886237332187st_nat(C,D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
            & 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),Aa)
              | 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_4104_atLeastatMost__psubset__iff,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(Aa,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
            & 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),Aa)
              | 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_4105_atLeastatMost__psubset__iff,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(Aa,B)),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Aa)
            & 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),Aa)
              | 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_4106_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_1: int] : aa_int_o(P3,X_1)
           => ? [X_12: int] : aa_int_o(P,X_12) ) ) ) ) ).

% plusinfinity
tff(fact_4107_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_1: int] : aa_int_o(P1,X_1)
           => ? [X_12: int] : aa_int_o(P,X_12) ) ) ) ) ).

% minusinfinity
tff(fact_4108_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_4109_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( insert_nat(Mb,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = set_or1269000886237332187st_nat(Mb,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_4110_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),set_or1269000886237332187st_nat(Mb,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_4111_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( set_or1269000886237332187st_nat(Mb,Nb) = insert_nat(Mb,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_4112_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_4113_unset__bit__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),Aa) = 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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_4114_unset__bit__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),Aa) = 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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_4115_Bolzano,axiom,
    ! [Aa: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => ( ! [A2: real,B3: real,C3: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A2),B3)
           => ( aa_real_o(aa_real_fun_real_o(P,B3),C3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),C3)
                 => aa_real_o(aa_real_fun_real_o(P,A2),C3) ) ) ) )
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
               => ? [D5: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
                    & ! [A2: real,B3: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(B3),A2)),D5) )
                       => aa_real_o(aa_real_fun_real_o(P,A2),B3) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,Aa),B) ) ) ) ).

% Bolzano
tff(fact_4116_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xb: real,Y3: 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(times_times_real(Xb),Z)),aa_real_real(times_times_real(Y3),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ).

% mult_le_cancel_iff1
tff(fact_4117_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xb: rat,Y3: 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(Y3),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3) ) ) ).

% mult_le_cancel_iff1
tff(fact_4118_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xb: int,Y3: 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,Y3),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3) ) ) ).

% mult_le_cancel_iff1
tff(fact_4119_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xb: real,Y3: 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(times_times_real(Z),Xb)),aa_real_real(times_times_real(Z),Y3))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ).

% mult_le_cancel_iff2
tff(fact_4120_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xb: rat,Y3: 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),Y3))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3) ) ) ).

% mult_le_cancel_iff2
tff(fact_4121_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xb: int,Y3: 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),Y3))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3) ) ) ).

% mult_le_cancel_iff2
tff(fact_4122_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(aa_num_num(bit1,Nb)),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_4123_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_4124_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_4125_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(aa_num_num(bit1,Nb)),unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_4126_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_4127_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_4128_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_4129_unset__bit__negative__int__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_se4203085406695923979it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_4130_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5055182867167087721od_nat(Mb,one) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_num_nat(numeral_numeral_nat,Mb)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_4131_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(Mb,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,Mb)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_4132_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique3479559517661332726nteger(Mb,one) = produc1086072967326762835nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_4133_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_4134_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_4135_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_4136_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_4137_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_4138_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_4139_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_4140_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(Nb),unique5055182867167087721od_nat(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_4141_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,Mb)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(Nb),unique5052692396658037445od_int(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_4142_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_4143_mult__less__iff1,axiom,
    ! [Z: real,Xb: real,Y3: 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(times_times_real(Xb),Z)),aa_real_real(times_times_real(Y3),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ) ).

% mult_less_iff1
tff(fact_4144_mult__less__iff1,axiom,
    ! [Z: rat,Xb: rat,Y3: 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(Y3),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3) ) ) ).

% mult_less_iff1
tff(fact_4145_mult__less__iff1,axiom,
    ! [Z: int,Xb: int,Y3: 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,Y3),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y3) ) ) ).

% mult_less_iff1
tff(fact_4146_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_4147_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_4148_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,Aa: 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,Aa),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))),Aa)),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_4149_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_4150_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A: set_nat,B2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A),Z)
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => ( ( A = B2 )
           => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A),Xb),X_12) ) ) ) ) ).

% obtain_set_succ
tff(fact_4151_obtain__set__pred,axiom,
    ! [Z: nat,Xb: nat,A: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A),Z)
       => ( aa_set_nat_o(finite_finite_nat,A)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A),Xb),X_12) ) ) ) ).

% obtain_set_pred
tff(fact_4152_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_4153_pred__none__empty,axiom,
    ! [Xs: set_nat,Aa: nat] :
      ( ~ ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Aa),X_12)
     => ( 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),Aa) ) ) ) ).

% pred_none_empty
tff(fact_4154_succ__none__empty,axiom,
    ! [Xs: set_nat,Aa: nat] :
      ( ~ ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Aa),X_12)
     => ( 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,Aa),X3) ) ) ) ).

% succ_none_empty
tff(fact_4155_old_Oprod_Oinject,axiom,
    ! [Aa: code_integer,B: $o,A4: code_integer,B5: $o] :
      ( ( produc6677183202524767010eger_o(Aa,(B)) = produc6677183202524767010eger_o(A4,(B5)) )
    <=> ( ( Aa = A4 )
        & ( (B)
        <=> (B5) ) ) ) ).

% old.prod.inject
tff(fact_4156_old_Oprod_Oinject,axiom,
    ! [Aa: num,B: num,A4: num,B5: num] :
      ( ( product_Pair_num_num(Aa,B) = product_Pair_num_num(A4,B5) )
    <=> ( ( Aa = A4 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_4157_old_Oprod_Oinject,axiom,
    ! [Aa: nat,B: num,A4: nat,B5: num] :
      ( ( product_Pair_nat_num(Aa,B) = product_Pair_nat_num(A4,B5) )
    <=> ( ( Aa = A4 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_4158_old_Oprod_Oinject,axiom,
    ! [Aa: nat,B: nat,A4: nat,B5: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Aa),B) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A4),B5) )
    <=> ( ( Aa = A4 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_4159_old_Oprod_Oinject,axiom,
    ! [Aa: int,B: int,A4: int,B5: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Aa),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B5) )
    <=> ( ( Aa = A4 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_4160_prod_Oinject,axiom,
    ! [X1: code_integer,X23: $o,Y1: code_integer,Y22: $o] :
      ( ( produc6677183202524767010eger_o(X1,(X23)) = produc6677183202524767010eger_o(Y1,(Y22)) )
    <=> ( ( X1 = Y1 )
        & ( (X23)
        <=> (Y22) ) ) ) ).

% prod.inject
tff(fact_4161_prod_Oinject,axiom,
    ! [X1: num,X23: num,Y1: num,Y22: num] :
      ( ( product_Pair_num_num(X1,X23) = product_Pair_num_num(Y1,Y22) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
tff(fact_4162_prod_Oinject,axiom,
    ! [X1: nat,X23: num,Y1: nat,Y22: num] :
      ( ( product_Pair_nat_num(X1,X23) = product_Pair_nat_num(Y1,Y22) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
tff(fact_4163_prod_Oinject,axiom,
    ! [X1: nat,X23: nat,Y1: nat,Y22: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X23) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
tff(fact_4164_prod_Oinject,axiom,
    ! [X1: int,X23: int,Y1: int,Y22: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
tff(fact_4165_mod__mod__trivial,axiom,
    ! [Aa: 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,Aa),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B) ).

% mod_mod_trivial
tff(fact_4166_mod__mod__trivial,axiom,
    ! [Aa: 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,Aa),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) ).

% mod_mod_trivial
tff(fact_4167_mod__mod__trivial,axiom,
    ! [Aa: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(Aa,B),B) = modulo364778990260209775nteger(Aa,B) ).

% mod_mod_trivial
tff(fact_4168_bits__mod__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% bits_mod_0
tff(fact_4169_bits__mod__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),Aa) = zero_zero_int ).

% bits_mod_0
tff(fact_4170_bits__mod__0,axiom,
    ! [Aa: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,Aa) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_4171_mod__add__self2,axiom,
    ! [Aa: 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),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B) ).

% mod_add_self2
tff(fact_4172_mod__add__self2,axiom,
    ! [Aa: 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),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) ).

% mod_add_self2
tff(fact_4173_mod__add__self2,axiom,
    ! [Aa: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(Aa,B),B) = modulo364778990260209775nteger(Aa,B) ).

% mod_add_self2
tff(fact_4174_mod__add__self1,axiom,
    ! [B: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B) ).

% mod_add_self1
tff(fact_4175_mod__add__self1,axiom,
    ! [B: int,Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) ).

% mod_add_self1
tff(fact_4176_mod__add__self1,axiom,
    ! [B: code_integer,Aa: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,Aa),B) = modulo364778990260209775nteger(Aa,B) ).

% mod_add_self1
tff(fact_4177_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_4178_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_4179_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_4180_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_4181_minus__mod__self2,axiom,
    ! [Aa: 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),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) ).

% minus_mod_self2
tff(fact_4182_minus__mod__self2,axiom,
    ! [Aa: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(Aa,B),B) = modulo364778990260209775nteger(Aa,B) ).

% minus_mod_self2
tff(fact_4183_mod__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = Mb ) ) ).

% mod_less
tff(fact_4184_mod__by__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_4185_mod__by__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_4186_mod__by__1,axiom,
    ! [Aa: code_integer] : modulo364778990260209775nteger(Aa,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_4187_bits__mod__by__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_4188_bits__mod__by__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_4189_bits__mod__by__1,axiom,
    ! [Aa: code_integer] : modulo364778990260209775nteger(Aa,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_4190_mod__mult__self2__is__0,axiom,
    ! [Aa: 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),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_4191_mod__mult__self2__is__0,axiom,
    ! [Aa: 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),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_4192_mod__mult__self2__is__0,axiom,
    ! [Aa: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_4193_mod__mult__self1__is__0,axiom,
    ! [B: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Aa)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_4194_mod__mult__self1__is__0,axiom,
    ! [B: int,Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Aa)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_4195_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,Aa: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Aa),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_4196_bits__mod__div__trivial,axiom,
    ! [Aa: 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,Aa),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_4197_bits__mod__div__trivial,axiom,
    ! [Aa: 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,Aa),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_4198_bits__mod__div__trivial,axiom,
    ! [Aa: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(Aa,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_4199_mod__div__trivial,axiom,
    ! [Aa: 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,Aa),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_4200_mod__div__trivial,axiom,
    ! [Aa: 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,Aa),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_4201_mod__div__trivial,axiom,
    ! [Aa: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(Aa,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_4202_mod__mult__self1,axiom,
    ! [Aa: 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),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,Aa),B) ).

% mod_mult_self1
tff(fact_4203_mod__mult__self1,axiom,
    ! [Aa: 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),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,Aa),B) ).

% mod_mult_self1
tff(fact_4204_mod__mult__self1,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(Aa,B) ).

% mod_mult_self1
tff(fact_4205_mod__mult__self2,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ).

% mod_mult_self2
tff(fact_4206_mod__mult__self2,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ).

% mod_mult_self2
tff(fact_4207_mod__mult__self2,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(Aa,B) ).

% mod_mult_self2
tff(fact_4208_mod__mult__self3,axiom,
    ! [C: nat,B: nat,Aa: 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)),Aa)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B) ).

% mod_mult_self3
tff(fact_4209_mod__mult__self3,axiom,
    ! [C: int,B: int,Aa: 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)),Aa)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) ).

% mod_mult_self3
tff(fact_4210_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,Aa: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),Aa),B) = modulo364778990260209775nteger(Aa,B) ).

% mod_mult_self3
tff(fact_4211_mod__mult__self4,axiom,
    ! [B: nat,C: nat,Aa: 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)),Aa)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B) ).

% mod_mult_self4
tff(fact_4212_mod__mult__self4,axiom,
    ! [B: int,C: int,Aa: 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)),Aa)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) ).

% mod_mult_self4
tff(fact_4213_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,Aa: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),Aa),B) = modulo364778990260209775nteger(Aa,B) ).

% mod_mult_self4
tff(fact_4214_infinite__Icc__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(Aa,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ).

% infinite_Icc_iff
tff(fact_4215_infinite__Icc__iff,axiom,
    ! [Aa: real,B: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(Aa,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ).

% infinite_Icc_iff
tff(fact_4216_mod__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_4217_Suc__mod__mult__self1,axiom,
    ! [Mb: 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,Mb),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,Mb)),Nb) ).

% Suc_mod_mult_self1
tff(fact_4218_Suc__mod__mult__self2,axiom,
    ! [Mb: 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,Mb),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,Mb)),Nb) ).

% Suc_mod_mult_self2
tff(fact_4219_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self3
tff(fact_4220_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self4
tff(fact_4221_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_4222_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_4223_bits__one__mod__two__eq__one,axiom,
    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_4224_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_4225_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_4226_one__mod__two__eq__one,axiom,
    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_4227_mod2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_4228_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_4229_not__mod__2__eq__1__eq__0,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4230_not__mod__2__eq__1__eq__0,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4231_not__mod__2__eq__1__eq__0,axiom,
    ! [Aa: code_integer] :
      ( ( modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4232_not__mod__2__eq__0__eq__1,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4233_not__mod__2__eq__0__eq__1,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4234_not__mod__2__eq__0__eq__1,axiom,
    ! [Aa: code_integer] :
      ( ( modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4235_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_4236_add__self__mod__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_4237_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_4238_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_4239_mod2__gr__0,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_4240_mod__mult__right__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_mult_right_eq
tff(fact_4241_mod__mult__right__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_mult_right_eq
tff(fact_4242_mod__mult__right__eq,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B),C) ).

% mod_mult_right_eq
tff(fact_4243_mod__mult__left__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_mult_left_eq
tff(fact_4244_mod__mult__left__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_mult_left_eq
tff(fact_4245_mod__mult__left__eq,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(Aa,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B),C) ).

% mod_mult_left_eq
tff(fact_4246_mult__mod__right,axiom,
    ! [C: nat,Aa: 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,Aa),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_4247_mult__mod__right,axiom,
    ! [C: int,Aa: 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,Aa),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ).

% mult_mod_right
tff(fact_4248_mult__mod__right,axiom,
    ! [C: code_integer,Aa: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),modulo364778990260209775nteger(Aa,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).

% mult_mod_right
tff(fact_4249_mod__mult__mult2,axiom,
    ! [Aa: 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),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,Aa),B)),C) ).

% mod_mult_mult2
tff(fact_4250_mod__mult__mult2,axiom,
    ! [Aa: 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),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,Aa),B)),C) ).

% mod_mult_mult2
tff(fact_4251_mod__mult__mult2,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(Aa,B)),C) ).

% mod_mult_mult2
tff(fact_4252_mod__mult__cong,axiom,
    ! [Aa: nat,C: nat,A4: nat,B: nat,B5: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),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,B5),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),B5)),C) ) ) ) ).

% mod_mult_cong
tff(fact_4253_mod__mult__cong,axiom,
    ! [Aa: int,C: int,A4: int,B: int,B5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),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,B5),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A4),B5)),C) ) ) ) ).

% mod_mult_cong
tff(fact_4254_mod__mult__cong,axiom,
    ! [Aa: code_integer,C: code_integer,A4: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(Aa,C) = modulo364778990260209775nteger(A4,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A4),B5),C) ) ) ) ).

% mod_mult_cong
tff(fact_4255_mod__mult__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_mult_eq
tff(fact_4256_mod__mult__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_mult_eq
tff(fact_4257_mod__mult__eq,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(Aa,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B),C) ).

% mod_mult_eq
tff(fact_4258_mod__add__right__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_add_right_eq
tff(fact_4259_mod__add__right__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_add_right_eq
tff(fact_4260_mod__add__right__eq,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(Aa,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(Aa,B),C) ).

% mod_add_right_eq
tff(fact_4261_mod__add__left__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_add_left_eq
tff(fact_4262_mod__add__left__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_add_left_eq
tff(fact_4263_mod__add__left__eq,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(Aa,B),C) ).

% mod_add_left_eq
tff(fact_4264_mod__add__cong,axiom,
    ! [Aa: nat,C: nat,A4: nat,B: nat,B5: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),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,B5),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_4265_mod__add__cong,axiom,
    ! [Aa: int,C: int,A4: int,B: int,B5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),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,B5),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_4266_mod__add__cong,axiom,
    ! [Aa: code_integer,C: code_integer,A4: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(Aa,C) = modulo364778990260209775nteger(A4,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(Aa,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A4,B5),C) ) ) ) ).

% mod_add_cong
tff(fact_4267_mod__add__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_add_eq
tff(fact_4268_mod__add__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_add_eq
tff(fact_4269_mod__add__eq,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(Aa,B),C) ).

% mod_add_eq
tff(fact_4270_mod__diff__right__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_diff_right_eq
tff(fact_4271_mod__diff__right__eq,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(Aa,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(Aa,B),C) ).

% mod_diff_right_eq
tff(fact_4272_mod__diff__left__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_diff_left_eq
tff(fact_4273_mod__diff__left__eq,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(Aa,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(Aa,B),C) ).

% mod_diff_left_eq
tff(fact_4274_mod__diff__cong,axiom,
    ! [Aa: int,C: int,A4: int,B: int,B5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),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,B5),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B5)),C) ) ) ) ).

% mod_diff_cong
tff(fact_4275_mod__diff__cong,axiom,
    ! [Aa: code_integer,C: code_integer,A4: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(Aa,C) = modulo364778990260209775nteger(A4,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(Aa,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A4,B5),C) ) ) ) ).

% mod_diff_cong
tff(fact_4276_mod__diff__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),B)),C) ).

% mod_diff_eq
tff(fact_4277_mod__diff__eq,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(Aa,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(Aa,B),C) ).

% mod_diff_eq
tff(fact_4278_power__mod,axiom,
    ! [Aa: 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,Aa),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,Aa),Nb)),B) ).

% power_mod
tff(fact_4279_power__mod,axiom,
    ! [Aa: 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,Aa),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,Aa),Nb)),B) ).

% power_mod
tff(fact_4280_power__mod,axiom,
    ! [Aa: code_integer,B: code_integer,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(Aa,B)),Nb),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb),B) ).

% power_mod
tff(fact_4281_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_4282_mod__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% mod_Suc_eq
tff(fact_4283_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_4284_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Nb) )
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% bounded_nat_set_is_finite
tff(fact_4285_finite__nat__set__iff__bounded,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
    <=> ? [M5: nat] :
        ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_4286_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
    <=> ? [M5: nat] :
        ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_4287_finite__list,axiom,
    ! [A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A ) ).

% finite_list
tff(fact_4288_finite__list,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A ) ).

% finite_list
tff(fact_4289_finite__list,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ? [Xs2: list_int] : set_int2(Xs2) = A ) ).

% finite_list
tff(fact_4290_finite__list,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A ) ).

% finite_list
tff(fact_4291_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_cw(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_4292_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F,N))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cx(F),U))) ) ).

% finite_less_ub
tff(fact_4293_finite__lists__length__eq,axiom,
    ! [A: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_cy(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4294_finite__lists__length__eq,axiom,
    ! [A: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_cz(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4295_finite__lists__length__eq,axiom,
    ! [A: set_VEBT_VEBT,Nb: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_da(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4296_finite__lists__length__eq,axiom,
    ! [A: set_o,Nb: nat] :
      ( finite_finite_o(A)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_db(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4297_finite__lists__length__eq,axiom,
    ! [A: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_dc(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4298_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(Aa,B)),Aa) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4299_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B)),Aa) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4300_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)),Aa) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4301_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,Aa: 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,Aa),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4302_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,Aa: 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,Aa),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4303_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(Aa,B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4304_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: 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,Mb))),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,Mb)),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_4305_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: 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,Mb))),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,Mb)),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_4306_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4307_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),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_4308_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),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_4309_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,one)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% cong_exp_iff_simps(4)
tff(fact_4310_mod__eq__self__iff__div__eq__0,axiom,
    ! [Aa: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B) = Aa )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4311_mod__eq__self__iff__div__eq__0,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) = Aa )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4312_mod__eq__self__iff__div__eq__0,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(Aa,B) = Aa )
    <=> ( divide6298287555418463151nteger(Aa,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4313_mod__eqE,axiom,
    ! [Aa: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
     => ~ ! [D3: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,C),D3)) ) ).

% mod_eqE
tff(fact_4314_mod__eqE,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(Aa,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D3: code_integer] : B != plus_p5714425477246183910nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D3)) ) ).

% mod_eqE
tff(fact_4315_div__add1__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_4316_div__add1__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_add1_eq
tff(fact_4317_div__add1__eq,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(Aa,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(Aa,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_4318_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))) ).

% mod_Suc
tff(fact_4319_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,Mb: 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,Mb),P2)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P2)) ) )
           => aa_nat_o(P,Mb) ) ) ) ) ).

% mod_induct
tff(fact_4320_mod__less__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_4321_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_4322_mod__eq__0D,axiom,
    ! [Mb: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),D) = zero_zero_nat )
     => ? [Q3: nat] : Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_4323_mod__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb),Mb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ).

% mod_if
tff(fact_4324_mod__geq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_4325_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y3: 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,Y3),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,Y3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_4326_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_4327_infinite__Icc,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(Aa,B)) ) ).

% infinite_Icc
tff(fact_4328_infinite__Icc,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(Aa,B)) ) ).

% infinite_Icc
tff(fact_4329_finite__lists__length__le,axiom,
    ! [A: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_dd(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_4330_finite__lists__length__le,axiom,
    ! [A: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_de(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_4331_finite__lists__length__le,axiom,
    ! [A: set_VEBT_VEBT,Nb: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_df(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_4332_finite__lists__length__le,axiom,
    ! [A: set_o,Nb: nat] :
      ( finite_finite_o(A)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_dg(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_4333_finite__lists__length__le,axiom,
    ! [A: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_dh(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_4334_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),B)
       => ( modulo364778990260209775nteger(Aa,B) = Aa ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4335_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B) = Aa ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4336_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) = Aa ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4337_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(Aa,B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4338_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,Aa: 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,Aa),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4339_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,Aa: 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,Aa),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4340_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_4341_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_4342_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4343_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_4344_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_4345_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_4346_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_4347_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_4348_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != 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_4349_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),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_4350_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),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_4351_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_4352_mult__div__mod__eq,axiom,
    ! [B: nat,Aa: 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,Aa),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B)) = Aa ).

% mult_div_mod_eq
tff(fact_4353_mult__div__mod__eq,axiom,
    ! [B: int,Aa: 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,Aa),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)) = Aa ).

% mult_div_mod_eq
tff(fact_4354_mult__div__mod__eq,axiom,
    ! [B: code_integer,Aa: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(Aa,B)),modulo364778990260209775nteger(Aa,B)) = Aa ).

% mult_div_mod_eq
tff(fact_4355_mod__mult__div__eq,axiom,
    ! [Aa: 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,Aa),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B))) = Aa ).

% mod_mult_div_eq
tff(fact_4356_mod__mult__div__eq,axiom,
    ! [Aa: 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,Aa),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B))) = Aa ).

% mod_mult_div_eq
tff(fact_4357_mod__mult__div__eq,axiom,
    ! [Aa: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(Aa,B))) = Aa ).

% mod_mult_div_eq
tff(fact_4358_mod__div__mult__eq,axiom,
    ! [Aa: 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,Aa),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)),B)) = Aa ).

% mod_div_mult_eq
tff(fact_4359_mod__div__mult__eq,axiom,
    ! [Aa: 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,Aa),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)),B)) = Aa ).

% mod_div_mult_eq
tff(fact_4360_mod__div__mult__eq,axiom,
    ! [Aa: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(Aa,B)),B)) = Aa ).

% mod_div_mult_eq
tff(fact_4361_div__mult__mod__eq,axiom,
    ! [Aa: 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,Aa),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B)) = Aa ).

% div_mult_mod_eq
tff(fact_4362_div__mult__mod__eq,axiom,
    ! [Aa: 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,Aa),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)) = Aa ).

% div_mult_mod_eq
tff(fact_4363_div__mult__mod__eq,axiom,
    ! [Aa: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(Aa,B)),B),modulo364778990260209775nteger(Aa,B)) = Aa ).

% div_mult_mod_eq
tff(fact_4364_mod__div__decomp,axiom,
    ! [Aa: nat,B: nat] : Aa = 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,Aa),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B)) ).

% mod_div_decomp
tff(fact_4365_mod__div__decomp,axiom,
    ! [Aa: int,B: int] : Aa = 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,Aa),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)) ).

% mod_div_decomp
tff(fact_4366_mod__div__decomp,axiom,
    ! [Aa: code_integer,B: code_integer] : Aa = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(Aa,B)),B),modulo364778990260209775nteger(Aa,B)) ).

% mod_div_decomp
tff(fact_4367_cancel__div__mod__rules_I1_J,axiom,
    ! [Aa: 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,Aa),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) ).

% cancel_div_mod_rules(1)
tff(fact_4368_cancel__div__mod__rules_I1_J,axiom,
    ! [Aa: 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,Aa),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) ).

% cancel_div_mod_rules(1)
tff(fact_4369_cancel__div__mod__rules_I1_J,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(Aa,B)),B),modulo364778990260209775nteger(Aa,B)),C) = plus_p5714425477246183910nteger(Aa,C) ).

% cancel_div_mod_rules(1)
tff(fact_4370_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,Aa: 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,Aa),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) ).

% cancel_div_mod_rules(2)
tff(fact_4371_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,Aa: 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,Aa),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) ).

% cancel_div_mod_rules(2)
tff(fact_4372_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,Aa: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(Aa,B)),modulo364778990260209775nteger(Aa,B)),C) = plus_p5714425477246183910nteger(Aa,C) ).

% cancel_div_mod_rules(2)
tff(fact_4373_div__mult1__eq,axiom,
    ! [Aa: 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,Aa),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),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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_4374_div__mult1__eq,axiom,
    ! [Aa: 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,Aa),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),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,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_mult1_eq
tff(fact_4375_div__mult1__eq,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_4376_minus__mult__div__eq__mod,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B) ).

% minus_mult_div_eq_mod
tff(fact_4377_minus__mult__div__eq__mod,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) ).

% minus_mult_div_eq_mod
tff(fact_4378_minus__mult__div__eq__mod,axiom,
    ! [Aa: code_integer,B: code_integer] : minus_8373710615458151222nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(Aa,B))) = modulo364778990260209775nteger(Aa,B) ).

% minus_mult_div_eq_mod
tff(fact_4379_minus__mod__eq__mult__div,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)) ).

% minus_mod_eq_mult_div
tff(fact_4380_minus__mod__eq__mult__div,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)) ).

% minus_mod_eq_mult_div
tff(fact_4381_minus__mod__eq__mult__div,axiom,
    ! [Aa: code_integer,B: code_integer] : minus_8373710615458151222nteger(Aa,modulo364778990260209775nteger(Aa,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(Aa,B)) ).

% minus_mod_eq_mult_div
tff(fact_4382_minus__mod__eq__div__mult,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4383_minus__mod__eq__div__mult,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4384_minus__mod__eq__div__mult,axiom,
    ! [Aa: code_integer,B: code_integer] : minus_8373710615458151222nteger(Aa,modulo364778990260209775nteger(Aa,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(Aa,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4385_minus__div__mult__eq__mod,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B) ).

% minus_div_mult_eq_mod
tff(fact_4386_minus__div__mult__eq__mod,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) ).

% minus_div_mult_eq_mod
tff(fact_4387_minus__div__mult__eq__mod,axiom,
    ! [Aa: code_integer,B: code_integer] : minus_8373710615458151222nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(Aa,B)),B)) = modulo364778990260209775nteger(Aa,B) ).

% minus_div_mult_eq_mod
tff(fact_4388_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: 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,Mb))),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_4389_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: 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,Mb))),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_4390_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != 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_4391_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: 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,Mb))),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_4392_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: 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,Mb))),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_4393_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != 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_4394_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: 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,Mb))),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,Mb)),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_4395_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: 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,Mb))),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,Mb)),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_4396_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4397_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_4398_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y3: 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,Y3),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xb)
       => ? [Q3: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_4399_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),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,Mb),Nb)
       => ~ ! [S2: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_4400_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),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),Mb)
       => ~ ! [S2: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_4401_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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,Mb),Nb)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% mod_mult2_eq
tff(fact_4402_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,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,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_4403_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_4404_modulo__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_4405_Pair__inject,axiom,
    ! [Aa: code_integer,B: $o,A4: code_integer,B5: $o] :
      ( ( produc6677183202524767010eger_o(Aa,(B)) = produc6677183202524767010eger_o(A4,(B5)) )
     => ~ ( ( Aa = A4 )
         => ( (B)
          <=> ~ (B5) ) ) ) ).

% Pair_inject
tff(fact_4406_Pair__inject,axiom,
    ! [Aa: num,B: num,A4: num,B5: num] :
      ( ( product_Pair_num_num(Aa,B) = product_Pair_num_num(A4,B5) )
     => ~ ( ( Aa = A4 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_4407_Pair__inject,axiom,
    ! [Aa: nat,B: num,A4: nat,B5: num] :
      ( ( product_Pair_nat_num(Aa,B) = product_Pair_nat_num(A4,B5) )
     => ~ ( ( Aa = A4 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_4408_Pair__inject,axiom,
    ! [Aa: nat,B: nat,A4: nat,B5: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Aa),B) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A4),B5) )
     => ~ ( ( Aa = A4 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_4409_Pair__inject,axiom,
    ! [Aa: int,B: int,A4: int,B5: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Aa),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B5) )
     => ~ ( ( Aa = A4 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_4410_prod__cases,axiom,
    ! [P: fun_Pr3290071492984375169er_o_o,P2: produc6271795597528267376eger_o] :
      ( ! [A2: code_integer,B3: $o] : aa_Pro5286387174248033258er_o_o(P,produc6677183202524767010eger_o(A2,(B3)))
     => aa_Pro5286387174248033258er_o_o(P,P2) ) ).

% prod_cases
tff(fact_4411_prod__cases,axiom,
    ! [P: fun_Pr6331313853612757382_num_o,P2: product_prod_num_num] :
      ( ! [A2: num,B3: num] : aa_Pro7685007914727650815_num_o(P,product_Pair_num_num(A2,B3))
     => aa_Pro7685007914727650815_num_o(P,P2) ) ).

% prod_cases
tff(fact_4412_prod__cases,axiom,
    ! [P: fun_Pr5555124106300582480_num_o,P2: product_prod_nat_num] :
      ( ! [A2: nat,B3: num] : aa_Pro6908818167415475913_num_o(P,product_Pair_nat_num(A2,B3))
     => aa_Pro6908818167415475913_num_o(P,P2) ) ).

% prod_cases
tff(fact_4413_prod__cases,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat] :
      ( ! [A2: nat,B3: nat] : aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B3))
     => aa_Pro6150637604445964947_nat_o(P,P2) ) ).

% prod_cases
tff(fact_4414_prod__cases,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,P2: product_prod_int_int] :
      ( ! [A2: int,B3: int] : aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B3))
     => aa_Pro4592068669816069595_int_o(P,P2) ) ).

% prod_cases
tff(fact_4415_surj__pair,axiom,
    ! [P2: produc6271795597528267376eger_o] :
    ? [X2: code_integer,Y: $o] : P2 = produc6677183202524767010eger_o(X2,(Y)) ).

% surj_pair
tff(fact_4416_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X2: num,Y: num] : P2 = product_Pair_num_num(X2,Y) ).

% surj_pair
tff(fact_4417_surj__pair,axiom,
    ! [P2: product_prod_nat_num] :
    ? [X2: nat,Y: num] : P2 = product_Pair_nat_num(X2,Y) ).

% surj_pair
tff(fact_4418_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X2: nat,Y: nat] : P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y) ).

% surj_pair
tff(fact_4419_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X2: int,Y: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y) ).

% surj_pair
tff(fact_4420_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc6271795597528267376eger_o] :
      ~ ! [A2: code_integer,B3: $o] : Y3 != produc6677183202524767010eger_o(A2,(B3)) ).

% old.prod.exhaust
tff(fact_4421_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_num_num] :
      ~ ! [A2: num,B3: num] : Y3 != product_Pair_num_num(A2,B3) ).

% old.prod.exhaust
tff(fact_4422_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_num] :
      ~ ! [A2: nat,B3: num] : Y3 != product_Pair_nat_num(A2,B3) ).

% old.prod.exhaust
tff(fact_4423_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ! [A2: nat,B3: nat] : Y3 != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B3) ).

% old.prod.exhaust
tff(fact_4424_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_int_int] :
      ~ ! [A2: int,B3: int] : Y3 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B3) ).

% old.prod.exhaust
tff(fact_4425_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N3),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4426_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_4427_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_4428_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : 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_4429_split__mod,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,Mb) )
        & ( ( Nb != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I5)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_4430_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,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,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_4431_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_4432_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% divmod_def
tff(fact_4433_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( modulo364778990260209775nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(Aa,B),C)),modulo364778990260209775nteger(Aa,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4434_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,Aa: 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,Aa),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,Aa),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4435_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,Aa: 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,Aa),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,Aa),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4436_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_4437_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_4438_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4439_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,Mb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4440_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,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,Mb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4441_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4442_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_4443_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))),Mb) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_4444_divmod__digit__0_I2_J,axiom,
    ! [B: nat,Aa: 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,Aa),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,Aa),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,Aa),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4445_divmod__digit__0_I2_J,axiom,
    ! [B: int,Aa: 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,Aa),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,Aa),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,Aa),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4446_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B)
       => ( modulo364778990260209775nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(Aa,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4447_bits__stable__imp__add__self,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Aa )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_4448_bits__stable__imp__add__self,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Aa )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_4449_bits__stable__imp__add__self,axiom,
    ! [Aa: code_integer] :
      ( ( divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = Aa )
     => ( plus_p5714425477246183910nteger(Aa,modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_4450_div__exp__mod__exp__eq,axiom,
    ! [Aa: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_4451_div__exp__mod__exp__eq,axiom,
    ! [Aa: int,Nb: nat,Mb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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))),Mb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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),Mb)))),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_4452_div__exp__mod__exp__eq,axiom,
    ! [Aa: code_integer,Nb: nat,Mb: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(Aa,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))),Mb)) = divide6298287555418463151nteger(modulo364778990260209775nteger(Aa,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),Mb))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_4453_divmod__digit__0_I1_J,axiom,
    ! [B: nat,Aa: 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,Aa),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,Aa),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,Aa),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4454_divmod__digit__0_I1_J,axiom,
    ! [B: int,Aa: 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,Aa),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,Aa),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,Aa),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4455_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(Aa,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))),divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(Aa,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4456_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4457_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),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,Aa),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),Mb)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4458_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,Aa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(Aa,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),Mb)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4459_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_4460_mod__double__modulus,axiom,
    ! [Mb: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Mb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
       => ( ( modulo364778990260209775nteger(Xb,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) = modulo364778990260209775nteger(Xb,Mb) )
          | ( modulo364778990260209775nteger(Xb,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Xb,Mb),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_4461_mod__double__modulus,axiom,
    ! [Mb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_4462_mod__double__modulus,axiom,
    ! [Mb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_4463_divmod__digit__1_I2_J,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(Aa,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4464_divmod__digit__1_I2_J,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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,Aa),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,Aa),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,Aa),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4465_divmod__digit__1_I2_J,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),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,Aa),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,Aa),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4466_unset__bit__Suc,axiom,
    ! [Nb: nat,Aa: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Nb),Aa) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,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))),bit_se8260200283734997820nteger(Nb,divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4467_unset__bit__Suc,axiom,
    ! [Nb: nat,Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,Nb)),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4468_unset__bit__Suc,axiom,
    ! [Nb: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4469_set__bit__Suc,axiom,
    ! [Nb: nat,Aa: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Nb),Aa) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,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))),bit_se2793503036327961859nteger(Nb,divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4470_set__bit__Suc,axiom,
    ! [Nb: nat,Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,Nb)),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4471_set__bit__Suc,axiom,
    ! [Nb: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4472_divmod__digit__1_I1_J,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(Aa,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4473_divmod__digit__1_I1_J,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( 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,Aa),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,Aa),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,Aa),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4474_divmod__digit__1_I1_J,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),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,Aa),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,Aa),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4475_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,Aa: 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(Aa,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))),Aa)),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_4476_verit__le__mono__div,axiom,
    ! [A: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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,A),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_4477_mod__exhaust__less__4,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_4478_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_di,K))) ).

% finite_Collect_le_nat
tff(fact_4479_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dj,K))) ).

% finite_Collect_less_nat
tff(fact_4480_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_dk(A))) ) ).

% finite_Collect_subsets
tff(fact_4481_finite__Collect__subsets,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_dl(A))) ) ).

% finite_Collect_subsets
tff(fact_4482_finite__Collect__subsets,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_dm(A))) ) ).

% finite_Collect_subsets
tff(fact_4483_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(aTP_Lamm_dn(Nb))) ) ).

% finite_roots_unity
tff(fact_4484_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_do,Nb))) ) ).

% finite_roots_unity
tff(fact_4485_finite__remove__induct,axiom,
    ! [B2: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B2)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [A5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A5)
             => ( ( A5 != bot_bo8194388402131092736T_VEBT )
               => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A5),B2)
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),A5)
                       => aa_set_VEBT_VEBT_o(P,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT))) )
                   => aa_set_VEBT_VEBT_o(P,A5) ) ) ) )
         => aa_set_VEBT_VEBT_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_4486_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( ( A5 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A5),B2)
                 => ( ! [X3: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X3),A5)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A5,insert_set_nat(X3,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A5) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_4487_finite__remove__induct,axiom,
    ! [B2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( ( A5 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A5),B2)
                 => ( ! [X3: complex] :
                        ( aa_set_complex_o(member_complex(X3),A5)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A5) ) ) ) )
         => aa_set_complex_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_4488_finite__remove__induct,axiom,
    ! [B2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ( A5 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A5),B2)
                 => ( ! [X3: real] :
                        ( aa_set_real_o(member_real(X3),A5)
                       => aa_set_real_o(P,minus_minus_set_real(A5,insert_real(X3,bot_bot_set_real))) )
                   => aa_set_real_o(P,A5) ) ) ) )
         => aa_set_real_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_4489_finite__remove__induct,axiom,
    ! [B2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ( A5 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A5),B2)
                 => ( ! [X3: nat] :
                        ( aa_set_nat_o(member_nat(X3),A5)
                       => aa_set_nat_o(P,minus_minus_set_nat(A5,insert_nat(X3,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A5) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_4490_finite__remove__induct,axiom,
    ! [B2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ( A5 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A5),B2)
                 => ( ! [X3: int] :
                        ( aa_set_int_o(member_int(X3),A5)
                       => aa_set_int_o(P,minus_minus_set_int(A5,insert_int(X3,bot_bot_set_int))) )
                   => aa_set_int_o(P,A5) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_4491_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y22: num] :
      ( ( aa_num_num(bit0,X23) = aa_num_num(bit0,Y22) )
    <=> ( X23 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_4492_verit__eq__simplify_I9_J,axiom,
    ! [X33: num,Y32: num] :
      ( ( aa_num_num(bit1,X33) = aa_num_num(bit1,Y32) )
    <=> ( X33 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_4493_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_dp(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4494_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
        | aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q)) )
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dq(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4495_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_dr(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4496_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_ds(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4497_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_dt(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4498_finite__Collect__disjI,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_du(P),Q)))
    <=> ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
        & finite2998713641127702882nt_int(collec213857154873943460nt_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_4499_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dv(P),Q)))
    <=> ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
        & aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_4500_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_dw(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_4501_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_dx(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_4502_finite__Collect__disjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dy(P),Q)))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        & aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_4503_finite__interval__int1,axiom,
    ! [Aa: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_dz(Aa),B))) ).

% finite_interval_int1
tff(fact_4504_finite__interval__int4,axiom,
    ! [Aa: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ea(Aa),B))) ).

% finite_interval_int4
tff(fact_4505_finite__interval__int2,axiom,
    ! [Aa: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_eb(Aa),B))) ).

% finite_interval_int2
tff(fact_4506_finite__interval__int3,axiom,
    ! [Aa: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ec(Aa),B))) ).

% finite_interval_int3
tff(fact_4507_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_4508_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_4509_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_4510_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_4511_zmod__le__nonneg__dividend,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),K)),Mb) ) ).

% zmod_le_nonneg_dividend
tff(fact_4512_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_4513_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_4514_zmod__eq__0__iff,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
    <=> ? [Q4: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_4515_zmod__eq__0D,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
     => ? [Q3: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_4516_finite__maxlen,axiom,
    ! [M6: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M6)
     => ? [N: nat] :
        ! [X3: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X3,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X3)),N) ) ) ).

% finite_maxlen
tff(fact_4517_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( finite_finite_list_o(M6)
     => ? [N: nat] :
        ! [X3: list_o] :
          ( member_list_o(X3,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X3)),N) ) ) ).

% finite_maxlen
tff(fact_4518_finite__maxlen,axiom,
    ! [M6: set_list_int] :
      ( finite3922522038869484883st_int(M6)
     => ? [N: nat] :
        ! [X3: list_int] :
          ( member_list_int(X3,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X3)),N) ) ) ).

% finite_maxlen
tff(fact_4519_div__mod__decomp__int,axiom,
    ! [A: int,Nb: 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),Nb)),Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Nb)) ).

% div_mod_decomp_int
tff(fact_4520_neg__mod__conj,axiom,
    ! [B: int,Aa: 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,Aa),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,Aa),B)) ) ) ).

% neg_mod_conj
tff(fact_4521_pos__mod__conj,axiom,
    ! [B: int,Aa: 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,Aa),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)),B) ) ) ).

% pos_mod_conj
tff(fact_4522_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_4523_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_4524_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_4525_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: set_int] : aa_set_int_o(ord_less_eq_set_int(Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_4526_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_4527_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_4528_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_4529_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_4530_verit__la__disequality,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Aa) ) ).

% verit_la_disequality
tff(fact_4531_verit__la__disequality,axiom,
    ! [Aa: num,B: num] :
      ( ( Aa = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),Aa) ) ).

% verit_la_disequality
tff(fact_4532_verit__la__disequality,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Aa) ) ).

% verit_la_disequality
tff(fact_4533_verit__la__disequality,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Aa) ) ).

% verit_la_disequality
tff(fact_4534_divmod__int__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_4535_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_4536_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_4537_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_4538_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_4539_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_4540_verit__la__generic,axiom,
    ! [Aa: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Xb)
      | ( Aa = Xb )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Aa) ) ).

% verit_la_generic
tff(fact_4541_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_4542_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_4543_verit__le__mono__div__int,axiom,
    ! [A: int,B2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( 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,A),Nb)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_4544_not__finite__existsD,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( ~ finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
     => ? [X_12: product_prod_int_int] : aa_Pro4592068669816069595_int_o(P,X_12) ) ).

% not_finite_existsD
tff(fact_4545_not__finite__existsD,axiom,
    ! [P: fun_set_nat_o] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
     => ? [X_12: set_nat] : aa_set_nat_o(P,X_12) ) ).

% not_finite_existsD
tff(fact_4546_not__finite__existsD,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ? [X_12: nat] : aa_nat_o(P,X_12) ) ).

% not_finite_existsD
tff(fact_4547_not__finite__existsD,axiom,
    ! [P: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(P))
     => ? [X_12: int] : aa_int_o(P,X_12) ) ).

% not_finite_existsD
tff(fact_4548_not__finite__existsD,axiom,
    ! [P: fun_complex_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
     => ? [X_12: complex] : aa_complex_o(P,X_12) ) ).

% not_finite_existsD
tff(fact_4549_pigeonhole__infinite__rel,axiom,
    ! [A: set_real,B2: set_nat,R: fun_real_fun_nat_o] :
      ( ~ aa_set_real_o(finite_finite_real,A)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B2)
                  & aa_nat_o(aa_real_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B2)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_ed(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4550_pigeonhole__infinite__rel,axiom,
    ! [A: set_real,B2: set_int,R: fun_real_fun_int_o] :
      ( ~ aa_set_real_o(finite_finite_real,A)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B2)
                  & aa_int_o(aa_real_fun_int_o(R,X2),Xa) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B2)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_ee(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4551_pigeonhole__infinite__rel,axiom,
    ! [A: set_real,B2: set_complex,R: fun_re7319081385389557872plex_o] :
      ( ~ aa_set_real_o(finite_finite_real,A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B2)
                  & aa_complex_o(aa_rea2382570592068293209plex_o(R,X2),Xa) ) )
         => ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),B2)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_ef(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4552_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B2: set_nat,R: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B2)
                  & aa_nat_o(aa_nat_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B2)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_eg(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4553_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B2: set_int,R: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B2)
                  & aa_int_o(aa_nat_fun_int_o(R,X2),Xa) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B2)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_eh(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4554_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B2: set_complex,R: fun_na1554462459066888140plex_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B2)
                  & aa_complex_o(aa_nat_fun_complex_o(R,X2),Xa) ) )
         => ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),B2)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_ei(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4555_pigeonhole__infinite__rel,axiom,
    ! [A: set_int,B2: set_nat,R: fun_int_fun_nat_o] :
      ( ~ aa_set_int_o(finite_finite_int,A)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B2)
                  & aa_nat_o(aa_int_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B2)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_ej(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4556_pigeonhole__infinite__rel,axiom,
    ! [A: set_int,B2: set_int,R: fun_int_fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B2)
                  & aa_int_o(aa_int_fun_int_o(R,X2),Xa) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B2)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ek(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4557_pigeonhole__infinite__rel,axiom,
    ! [A: set_int,B2: set_complex,R: fun_in5065666251602127472plex_o] :
      ( ~ aa_set_int_o(finite_finite_int,A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B2)
                  & aa_complex_o(aa_int_fun_complex_o(R,X2),Xa) ) )
         => ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),B2)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_el(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4558_pigeonhole__infinite__rel,axiom,
    ! [A: set_complex,B2: set_nat,R: fun_co5676208753217392844_nat_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B2)
                  & aa_nat_o(aa_complex_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B2)
              & ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_em(A),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4559_int__mod__pos__eq,axiom,
    ! [Aa: int,B: int,Q2: int,R2: int] :
      ( ( Aa = 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,Aa),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_4560_int__mod__neg__eq,axiom,
    ! [Aa: int,B: int,Q2: int,R2: int] :
      ( ( Aa = 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,Aa),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_4561_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)
         => ! [I5: 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),I5)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I5: 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),I5)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_4562_zmod__zmult2__eq,axiom,
    ! [C: int,Aa: 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,Aa),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,Aa),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_4563_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))
      <=> ! [I5: 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),I5)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I5),J3) ) ) ) ).

% split_neg_lemma
tff(fact_4564_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))
      <=> ! [I5: 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),I5)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I5),J3) ) ) ) ).

% split_pos_lemma
tff(fact_4565_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A4: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B5),A4)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A4),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4566_verit__comp__simplify1_I3_J,axiom,
    ! [B5: rat,A4: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B5),A4)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4567_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A4: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B5),A4)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4568_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A4: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B5),A4)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4569_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A4: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),A4)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4570_verit__sum__simplify,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),zero_zero_complex) = Aa ).

% verit_sum_simplify
tff(fact_4571_verit__sum__simplify,axiom,
    ! [Aa: real] : aa_real_real(plus_plus_real(Aa),zero_zero_real) = Aa ).

% verit_sum_simplify
tff(fact_4572_verit__sum__simplify,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(Aa),zero_zero_rat) = Aa ).

% verit_sum_simplify
tff(fact_4573_verit__sum__simplify,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),zero_zero_nat) = Aa ).

% verit_sum_simplify
tff(fact_4574_verit__sum__simplify,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),zero_zero_int) = Aa ).

% verit_sum_simplify
tff(fact_4575_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one != aa_num_num(bit0,X23) ).

% verit_eq_simplify(10)
tff(fact_4576_finite__has__minimal2,axiom,
    ! [A: set_real,Aa: real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_real_o(member_real(Aa),A)
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Aa)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4577_finite__has__minimal2,axiom,
    ! [A: set_set_nat,Aa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A)
     => ( aa_set_set_nat_o(member_set_nat(Aa),A)
       => ? [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A)
            & aa_set_nat_o(ord_less_eq_set_nat(X2),Aa)
            & ! [Xa: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa),A)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4578_finite__has__minimal2,axiom,
    ! [A: set_set_int,Aa: set_int] :
      ( finite6197958912794628473et_int(A)
     => ( member_set_int(Aa,A)
       => ? [X2: set_int] :
            ( member_set_int(X2,A)
            & aa_set_int_o(ord_less_eq_set_int(X2),Aa)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4579_finite__has__minimal2,axiom,
    ! [A: set_rat,Aa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(member_rat(Aa),A)
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Aa)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4580_finite__has__minimal2,axiom,
    ! [A: set_num,Aa: num] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( aa_set_num_o(member_num(Aa),A)
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Aa)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4581_finite__has__minimal2,axiom,
    ! [A: set_nat,Aa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(Aa),A)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Aa)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4582_finite__has__minimal2,axiom,
    ! [A: set_int,Aa: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(Aa),A)
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Aa)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4583_finite__has__maximal2,axiom,
    ! [A: set_real,Aa: real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_real_o(member_real(Aa),A)
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4584_finite__has__maximal2,axiom,
    ! [A: set_set_nat,Aa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A)
     => ( aa_set_set_nat_o(member_set_nat(Aa),A)
       => ? [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A)
            & aa_set_nat_o(ord_less_eq_set_nat(Aa),X2)
            & ! [Xa: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa),A)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4585_finite__has__maximal2,axiom,
    ! [A: set_set_int,Aa: set_int] :
      ( finite6197958912794628473et_int(A)
     => ( member_set_int(Aa,A)
       => ? [X2: set_int] :
            ( member_set_int(X2,A)
            & aa_set_int_o(ord_less_eq_set_int(Aa),X2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A)
               => ( aa_set_int_o(ord_less_eq_set_int(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4586_finite__has__maximal2,axiom,
    ! [A: set_rat,Aa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(member_rat(Aa),A)
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),X2)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4587_finite__has__maximal2,axiom,
    ! [A: set_num,Aa: num] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( aa_set_num_o(member_num(Aa),A)
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),X2)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4588_finite__has__maximal2,axiom,
    ! [A: set_nat,Aa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(Aa),A)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),X2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4589_finite__has__maximal2,axiom,
    ! [A: set_int,Aa: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(Aa),A)
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),X2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4590_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_4591_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] : one != aa_num_num(bit1,X33) ).

% verit_eq_simplify(12)
tff(fact_4592_finite__subset,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => aa_set_nat_o(finite_finite_nat,A) ) ) ).

% finite_subset
tff(fact_4593_finite__subset,axiom,
    ! [A: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => aa_set_complex_o(finite3207457112153483333omplex,A) ) ) ).

% finite_subset
tff(fact_4594_finite__subset,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => aa_set_int_o(finite_finite_int,A) ) ) ).

% finite_subset
tff(fact_4595_infinite__super,axiom,
    ! [S3: set_nat,T2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
     => ( ~ aa_set_nat_o(finite_finite_nat,S3)
       => ~ aa_set_nat_o(finite_finite_nat,T2) ) ) ).

% infinite_super
tff(fact_4596_infinite__super,axiom,
    ! [S3: set_complex,T2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S3)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T2) ) ) ).

% infinite_super
tff(fact_4597_infinite__super,axiom,
    ! [S3: set_int,T2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
     => ( ~ aa_set_int_o(finite_finite_int,S3)
       => ~ aa_set_int_o(finite_finite_int,T2) ) ) ).

% infinite_super
tff(fact_4598_rev__finite__subset,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
       => aa_set_nat_o(finite_finite_nat,A) ) ) ).

% rev_finite_subset
tff(fact_4599_rev__finite__subset,axiom,
    ! [B2: set_complex,A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
       => aa_set_complex_o(finite3207457112153483333omplex,A) ) ) ).

% rev_finite_subset
tff(fact_4600_rev__finite__subset,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A),B2)
       => aa_set_int_o(finite_finite_int,A) ) ) ).

% rev_finite_subset
tff(fact_4601_pos__zmod__mult__2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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)) = 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))) ) ) ).

% pos_zmod_mult_2
tff(fact_4602_max__def__raw,axiom,
    ! [X3: extended_enat,Xa: extended_enat] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X3),Xa) = $ite(ord_le2932123472753598470d_enat(X3,Xa),Xa,X3) ).

% max_def_raw
tff(fact_4603_max__def__raw,axiom,
    ! [X3: code_integer,Xa: code_integer] :
      ord_max_Code_integer(X3,Xa) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4604_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_4605_max__def__raw,axiom,
    ! [X3: rat,Xa: rat] :
      aa_rat_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_4606_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(aa_num_fun_num_o(ord_less_eq_num,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4607_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_4608_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_4609_neg__zmod__mult__2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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))),Aa)) = 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)),Aa))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_4610_finite__has__maximal,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4611_finite__has__maximal,axiom,
    ! [A: set_set_int] :
      ( finite6197958912794628473et_int(A)
     => ( ( A != bot_bot_set_set_int )
       => ? [X2: set_int] :
            ( member_set_int(X2,A)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A)
               => ( aa_set_int_o(ord_less_eq_set_int(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4612_finite__has__maximal,axiom,
    ! [A: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4613_finite__has__maximal,axiom,
    ! [A: set_num] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( ( A != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4614_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4615_finite__has__maximal,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4616_finite__has__minimal,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4617_finite__has__minimal,axiom,
    ! [A: set_set_int] :
      ( finite6197958912794628473et_int(A)
     => ( ( A != bot_bot_set_set_int )
       => ? [X2: set_int] :
            ( member_set_int(X2,A)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4618_finite__has__minimal,axiom,
    ! [A: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4619_finite__has__minimal,axiom,
    ! [A: set_num] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( ( A != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4620_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4621_finite__has__minimal,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4622_finite__subset__induct,axiom,
    ! [F3: set_VEBT_VEBT,A: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F3),A)
       => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
         => ( ! [A2: vEBT_VEBT,F4: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F4)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),A)
                 => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),F4)
                   => ( aa_set_VEBT_VEBT_o(P,F4)
                     => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(A2,F4)) ) ) ) )
           => aa_set_VEBT_VEBT_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4623_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A2: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( aa_set_set_nat_o(member_set_nat(A2),A)
                 => ( ~ aa_set_set_nat_o(member_set_nat(A2),F4)
                   => ( aa_set_set_nat_o(P,F4)
                     => aa_set_set_nat_o(P,insert_set_nat(A2,F4)) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4624_finite__subset__induct,axiom,
    ! [F3: set_complex,A: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F3),A)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A2: complex,F4: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F4)
               => ( aa_set_complex_o(member_complex(A2),A)
                 => ( ~ aa_set_complex_o(member_complex(A2),F4)
                   => ( aa_set_complex_o(P,F4)
                     => aa_set_complex_o(P,insert_complex(A2,F4)) ) ) ) )
           => aa_set_complex_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4625_finite__subset__induct,axiom,
    ! [F3: set_nat,A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F3),A)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A2: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( aa_set_nat_o(member_nat(A2),A)
                 => ( ~ aa_set_nat_o(member_nat(A2),F4)
                   => ( aa_set_nat_o(P,F4)
                     => aa_set_nat_o(P,insert_nat(A2,F4)) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4626_finite__subset__induct,axiom,
    ! [F3: set_real,A: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(ord_less_eq_set_real(F3),A)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A2: real,F4: set_real] :
                ( aa_set_real_o(finite_finite_real,F4)
               => ( aa_set_real_o(member_real(A2),A)
                 => ( ~ aa_set_real_o(member_real(A2),F4)
                   => ( aa_set_real_o(P,F4)
                     => aa_set_real_o(P,insert_real(A2,F4)) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4627_finite__subset__induct,axiom,
    ! [F3: set_int,A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(ord_less_eq_set_int(F3),A)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A2: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( aa_set_int_o(member_int(A2),A)
                 => ( ~ aa_set_int_o(member_int(A2),F4)
                   => ( aa_set_int_o(P,F4)
                     => aa_set_int_o(P,insert_int(A2,F4)) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4628_finite__subset__induct_H,axiom,
    ! [F3: set_VEBT_VEBT,A: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F3),A)
       => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
         => ( ! [A2: vEBT_VEBT,F4: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F4)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),A)
                 => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F4),A)
                   => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),F4)
                     => ( aa_set_VEBT_VEBT_o(P,F4)
                       => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(A2,F4)) ) ) ) ) )
           => aa_set_VEBT_VEBT_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4629_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A2: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( aa_set_set_nat_o(member_set_nat(A2),A)
                 => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F4),A)
                   => ( ~ aa_set_set_nat_o(member_set_nat(A2),F4)
                     => ( aa_set_set_nat_o(P,F4)
                       => aa_set_set_nat_o(P,insert_set_nat(A2,F4)) ) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4630_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F3),A)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A2: complex,F4: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F4)
               => ( aa_set_complex_o(member_complex(A2),A)
                 => ( aa_set_complex_o(ord_le211207098394363844omplex(F4),A)
                   => ( ~ aa_set_complex_o(member_complex(A2),F4)
                     => ( aa_set_complex_o(P,F4)
                       => aa_set_complex_o(P,insert_complex(A2,F4)) ) ) ) ) )
           => aa_set_complex_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4631_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F3),A)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A2: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( aa_set_nat_o(member_nat(A2),A)
                 => ( aa_set_nat_o(ord_less_eq_set_nat(F4),A)
                   => ( ~ aa_set_nat_o(member_nat(A2),F4)
                     => ( aa_set_nat_o(P,F4)
                       => aa_set_nat_o(P,insert_nat(A2,F4)) ) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4632_finite__subset__induct_H,axiom,
    ! [F3: set_real,A: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(ord_less_eq_set_real(F3),A)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A2: real,F4: set_real] :
                ( aa_set_real_o(finite_finite_real,F4)
               => ( aa_set_real_o(member_real(A2),A)
                 => ( aa_set_real_o(ord_less_eq_set_real(F4),A)
                   => ( ~ aa_set_real_o(member_real(A2),F4)
                     => ( aa_set_real_o(P,F4)
                       => aa_set_real_o(P,insert_real(A2,F4)) ) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4633_finite__subset__induct_H,axiom,
    ! [F3: set_int,A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(ord_less_eq_set_int(F3),A)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A2: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( aa_set_int_o(member_int(A2),A)
                 => ( aa_set_int_o(ord_less_eq_set_int(F4),A)
                   => ( ~ aa_set_int_o(member_int(A2),F4)
                     => ( aa_set_int_o(P,F4)
                       => aa_set_int_o(P,insert_int(A2,F4)) ) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4634_div__less__mono,axiom,
    ! [A: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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,A),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),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,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_4635_div__mod__decomp,axiom,
    ! [A: nat,Nb: 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),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Nb)) ).

% div_mod_decomp
tff(fact_4636_remove__induct,axiom,
    ! [P: fun_set_VEBT_VEBT_o,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
     => ( ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B2)
         => aa_set_VEBT_VEBT_o(P,B2) )
       => ( ! [A5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A5)
             => ( ( A5 != bot_bo8194388402131092736T_VEBT )
               => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A5),B2)
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),A5)
                       => aa_set_VEBT_VEBT_o(P,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT))) )
                   => aa_set_VEBT_VEBT_o(P,A5) ) ) ) )
         => aa_set_VEBT_VEBT_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_4637_remove__induct,axiom,
    ! [P: fun_set_set_nat_o,B2: set_set_nat] :
      ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
     => ( ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
         => aa_set_set_nat_o(P,B2) )
       => ( ! [A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( ( A5 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A5),B2)
                 => ( ! [X3: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X3),A5)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A5,insert_set_nat(X3,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A5) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_4638_remove__induct,axiom,
    ! [P: fun_set_complex_o,B2: set_complex] :
      ( aa_set_complex_o(P,bot_bot_set_complex)
     => ( ( ~ aa_set_complex_o(finite3207457112153483333omplex,B2)
         => aa_set_complex_o(P,B2) )
       => ( ! [A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( ( A5 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A5),B2)
                 => ( ! [X3: complex] :
                        ( aa_set_complex_o(member_complex(X3),A5)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A5) ) ) ) )
         => aa_set_complex_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_4639_remove__induct,axiom,
    ! [P: fun_set_real_o,B2: set_real] :
      ( aa_set_real_o(P,bot_bot_set_real)
     => ( ( ~ aa_set_real_o(finite_finite_real,B2)
         => aa_set_real_o(P,B2) )
       => ( ! [A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ( A5 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A5),B2)
                 => ( ! [X3: real] :
                        ( aa_set_real_o(member_real(X3),A5)
                       => aa_set_real_o(P,minus_minus_set_real(A5,insert_real(X3,bot_bot_set_real))) )
                   => aa_set_real_o(P,A5) ) ) ) )
         => aa_set_real_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_4640_remove__induct,axiom,
    ! [P: fun_set_nat_o,B2: set_nat] :
      ( aa_set_nat_o(P,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B2)
         => aa_set_nat_o(P,B2) )
       => ( ! [A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ( A5 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A5),B2)
                 => ( ! [X3: nat] :
                        ( aa_set_nat_o(member_nat(X3),A5)
                       => aa_set_nat_o(P,minus_minus_set_nat(A5,insert_nat(X3,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A5) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_4641_remove__induct,axiom,
    ! [P: fun_set_int_o,B2: set_int] :
      ( aa_set_int_o(P,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B2)
         => aa_set_int_o(P,B2) )
       => ( ! [A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ( A5 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A5),B2)
                 => ( ! [X3: int] :
                        ( aa_set_int_o(member_int(X3),A5)
                       => aa_set_int_o(P,minus_minus_set_int(A5,insert_int(X3,bot_bot_set_int))) )
                   => aa_set_int_o(P,A5) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_4642_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4643_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_en(Nb),C))) ) ).

% finite_nth_roots
tff(fact_4644_flip__bit__Suc,axiom,
    ! [Nb: nat,Aa: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Nb),Aa) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,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))),bit_se1345352211410354436nteger(Nb,divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4645_flip__bit__Suc,axiom,
    ! [Nb: nat,Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,Nb)),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4646_flip__bit__Suc,axiom,
    ! [Nb: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4647_finite__linorder__max__induct,axiom,
    ! [A: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B3: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),A5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B3) )
               => ( aa_set_real_o(P,A5)
                 => aa_set_real_o(P,insert_real(B3,A5)) ) ) )
         => aa_set_real_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4648_finite__linorder__max__induct,axiom,
    ! [A: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B3: rat,A5: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A5)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),B3) )
               => ( aa_set_rat_o(P,A5)
                 => aa_set_rat_o(P,insert_rat(B3,A5)) ) ) )
         => aa_set_rat_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4649_finite__linorder__max__induct,axiom,
    ! [A: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B3: num,A5: set_num] :
              ( aa_set_num_o(finite_finite_num,A5)
             => ( ! [X3: num] :
                    ( aa_set_num_o(member_num(X3),A5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),B3) )
               => ( aa_set_num_o(P,A5)
                 => aa_set_num_o(P,insert_num(B3,A5)) ) ) )
         => aa_set_num_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4650_finite__linorder__max__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B3: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),A5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),B3) )
               => ( aa_set_nat_o(P,A5)
                 => aa_set_nat_o(P,insert_nat(B3,A5)) ) ) )
         => aa_set_nat_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4651_finite__linorder__max__induct,axiom,
    ! [A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B3: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),B3) )
               => ( aa_set_int_o(P,A5)
                 => aa_set_int_o(P,insert_int(B3,A5)) ) ) )
         => aa_set_int_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4652_finite__linorder__min__induct,axiom,
    ! [A: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B3: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),A5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B3),X3) )
               => ( aa_set_real_o(P,A5)
                 => aa_set_real_o(P,insert_real(B3,A5)) ) ) )
         => aa_set_real_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4653_finite__linorder__min__induct,axiom,
    ! [A: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B3: rat,A5: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A5)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B3),X3) )
               => ( aa_set_rat_o(P,A5)
                 => aa_set_rat_o(P,insert_rat(B3,A5)) ) ) )
         => aa_set_rat_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4654_finite__linorder__min__induct,axiom,
    ! [A: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B3: num,A5: set_num] :
              ( aa_set_num_o(finite_finite_num,A5)
             => ( ! [X3: num] :
                    ( aa_set_num_o(member_num(X3),A5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B3),X3) )
               => ( aa_set_num_o(P,A5)
                 => aa_set_num_o(P,insert_num(B3,A5)) ) ) )
         => aa_set_num_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4655_finite__linorder__min__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B3: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),A5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),X3) )
               => ( aa_set_nat_o(P,A5)
                 => aa_set_nat_o(P,insert_nat(B3,A5)) ) ) )
         => aa_set_nat_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4656_finite__linorder__min__induct,axiom,
    ! [A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B3: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B3),X3) )
               => ( aa_set_int_o(P,A5)
                 => aa_set_int_o(P,insert_int(B3,A5)) ) ) )
         => aa_set_int_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4657_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X2: vEBT_VEBT,S4: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
             => ( ! [Y2: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y2),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,Y2)),aa_VEBT_VEBT_rat(F,X2)) )
               => ( aa_set_VEBT_VEBT_o(P,S4)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X2,S4)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4658_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_o,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,S4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S4)
             => ( ! [Y2: complex] :
                    ( aa_set_complex_o(member_complex(Y2),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Y2)),aa_complex_rat(F,X2)) )
               => ( aa_set_complex_o(P,S4)
                 => aa_set_complex_o(P,insert_complex(X2,S4)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4659_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S4)
             => ( ! [Y2: nat] :
                    ( aa_set_nat_o(member_nat(Y2),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y2)),aa_nat_rat(F,X2)) )
               => ( aa_set_nat_o(P,S4)
                 => aa_set_nat_o(P,insert_nat(X2,S4)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4660_finite__ranking__induct,axiom,
    ! [S3: set_int,P: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,S4: set_int] :
              ( aa_set_int_o(finite_finite_int,S4)
             => ( ! [Y2: int] :
                    ( aa_set_int_o(member_int(Y2),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y2)),aa_int_rat(F,X2)) )
               => ( aa_set_int_o(P,S4)
                 => aa_set_int_o(P,insert_int(X2,S4)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4661_finite__ranking__induct,axiom,
    ! [S3: set_real,P: fun_set_real_o,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X2: real,S4: set_real] :
              ( aa_set_real_o(finite_finite_real,S4)
             => ( ! [Y2: real] :
                    ( aa_set_real_o(member_real(Y2),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Y2)),aa_real_rat(F,X2)) )
               => ( aa_set_real_o(P,S4)
                 => aa_set_real_o(P,insert_real(X2,S4)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4662_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_num] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X2: vEBT_VEBT,S4: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
             => ( ! [Y2: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y2),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_VEBT_VEBT_num(F,Y2)),aa_VEBT_VEBT_num(F,X2)) )
               => ( aa_set_VEBT_VEBT_o(P,S4)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X2,S4)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4663_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_o,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,S4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S4)
             => ( ! [Y2: complex] :
                    ( aa_set_complex_o(member_complex(Y2),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,Y2)),aa_complex_num(F,X2)) )
               => ( aa_set_complex_o(P,S4)
                 => aa_set_complex_o(P,insert_complex(X2,S4)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4664_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: fun_set_nat_o,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S4)
             => ( ! [Y2: nat] :
                    ( aa_set_nat_o(member_nat(Y2),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Y2)),aa_nat_num(F,X2)) )
               => ( aa_set_nat_o(P,S4)
                 => aa_set_nat_o(P,insert_nat(X2,S4)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4665_finite__ranking__induct,axiom,
    ! [S3: set_int,P: fun_set_int_o,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,S4: set_int] :
              ( aa_set_int_o(finite_finite_int,S4)
             => ( ! [Y2: int] :
                    ( aa_set_int_o(member_int(Y2),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,Y2)),aa_int_num(F,X2)) )
               => ( aa_set_int_o(P,S4)
                 => aa_set_int_o(P,insert_int(X2,S4)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4666_finite__ranking__induct,axiom,
    ! [S3: set_real,P: fun_set_real_o,F: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X2: real,S4: set_real] :
              ( aa_set_real_o(finite_finite_real,S4)
             => ( ! [Y2: real] :
                    ( aa_set_real_o(member_real(Y2),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,Y2)),aa_real_num(F,X2)) )
               => ( aa_set_real_o(P,S4)
                 => aa_set_real_o(P,insert_real(X2,S4)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4667_product__nth,axiom,
    ! [Nb: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),Nb) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_4668_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr8326237132889035090at_num(product_nat_num(Xs,Ys),Nb) = product_Pair_nat_num(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_4669_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,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_size_list_nat(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr7617993195940197384at_nat(product_nat_nat(Xs,Ys),Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_4670_product__nth,axiom,
    ! [Nb: nat,Xs: list_Code_integer,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_s3445333598471063425nteger(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr8522763379788166057eger_o(produc3607205314601156340eger_o(Xs,Ys),Nb) = produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(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_4671_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_4672_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) = 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_4673_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),Nb) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_4674_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,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_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr6777367263587873994T_VEBT(product_o_VEBT_VEBT(Xs,Ys),Nb) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(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_4675_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,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_size_list_o(Xs)),size_size_list_o(Ys)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys),Nb) = product_Pair_o_o(aa_nat_o(nth_o(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_4676_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys),Nb) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_4677_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_4678_flip__bit__negative__int__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_se2159334234014336723it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_4679_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4680_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_4681_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4682_length__product,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT] : size_s4313452262239582901T_VEBT(product_o_VEBT_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4683_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_4684_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_4685_length__product,axiom,
    ! [Xs: list_int,Ys: list_VEBT_VEBT] : size_s6639371672096860321T_VEBT(produc662631939642741121T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4686_length__product,axiom,
    ! [Xs: list_int,Ys: list_o] : size_s4246224855604898693_int_o(product_int_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_4687_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] : size_s5157815400016825771nt_int(product_int_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4688_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),S3)
            & ~ ? [Xa: real] :
                  ( aa_set_real_o(member_real(Xa),S3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4689_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S3)
     => ( ( S3 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),S3)
            & ~ ? [Xa: rat] :
                  ( aa_set_rat_o(member_rat(Xa),S3)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4690_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( aa_set_num_o(finite_finite_num,S3)
     => ( ( S3 != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),S3)
            & ~ ? [Xa: num] :
                  ( aa_set_num_o(member_num(Xa),S3)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4691_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),S3)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S3)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4692_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),S3)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),S3)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4693_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_4694_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_4695_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(aa_num_fun_num_o(ord_less_num,X2),Xa) ) )
       => ~ aa_set_num_o(finite_finite_num,X5) ) ) ).

% infinite_growing
tff(fact_4696_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_4697_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_4698_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_complex,Y3: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_eo(I6),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_eo(I6),Y3)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_ep(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4699_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_complex,Y3: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_eq(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_eq(I6),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_er(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4700_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_complex,Y3: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_es(I6),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_es(I6),Y3)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_et(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4701_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_complex,Y3: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_eu(I6),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_eu(I6),Y3)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_ev(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4702_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_real,Y3: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_ew(I6),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_ew(I6),Y3)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ex(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4703_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_real,Y3: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ey(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ey(I6),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_ez(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4704_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_real,Y3: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_fa(I6),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_fa(I6),Y3)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_fb(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4705_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,Y3: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_fc(I6),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_fc(I6),Y3)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_fd(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4706_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_rat,Y3: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fe(I6),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fe(I6),Y3)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_ff(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4707_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,Y3: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_fg(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_fg(I6),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_fh(I6),Xb),Y3))) ) ) ).

% prod.finite_Collect_op
tff(fact_4708_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_complex,Y3: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_fi(I6),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_fi(I6),Y3)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_fj(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4709_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_complex,Y3: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_fk(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_fk(I6),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_fl(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4710_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_complex,Y3: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_fm(I6),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_fm(I6),Y3)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_fn(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4711_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_complex,Y3: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_fo(I6),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_fo(I6),Y3)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_fp(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4712_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_real,Y3: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_fq(I6),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_fq(I6),Y3)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_fr(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4713_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_real,Y3: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_fs(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_fs(I6),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_ft(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4714_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_real,Y3: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_fu(I6),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_fu(I6),Y3)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_fv(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4715_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,Y3: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_fw(I6),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_fw(I6),Y3)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_fx(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4716_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_rat,Y3: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fy(I6),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fy(I6),Y3)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_fz(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4717_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,Y3: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ga(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ga(I6),Y3)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_gb(I6),Xb),Y3))) ) ) ).

% sum.finite_Collect_op
tff(fact_4718_dbl__inc__simps_I3_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_4719_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_4720_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_4721_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_4722_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [Aa: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Leaf((Aa),(B))) = zero_zero_nat ).

% VEBT_internal.height.simps(1)
tff(fact_4723_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_4724_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_4725_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_4726_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_4727_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_4728_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_4729_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_4730_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_4731_dbl__inc__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,Xb) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Xb)),one_one_complex) ).

% dbl_inc_def
tff(fact_4732_dbl__inc__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu8295874005876285629c_real,Xb) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(Xb),Xb)),one_one_real) ).

% dbl_inc_def
tff(fact_4733_dbl__inc__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,Xb) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).

% dbl_inc_def
tff(fact_4734_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_4735_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A2: $o,B3: $o] : Xb != vEBT_Leaf((A2),(B3))
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb != vEBT_Node(Uu2,Deg,TreeList,Summary) ) ).

% VEBT_internal.height.cases
tff(fact_4736_divmod__BitM__2__eq,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(aa_num_num(bitM,Mb),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_4737_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [M2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M2),zero_zero_nat)
     => ( ! [M2: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M2),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_4738_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_4739_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4740_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4741_dbl__simps_I3_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4742_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4743_even__succ__mod__exp,axiom,
    ! [Aa: 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)
     => ( 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),Aa)),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,Aa),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_4744_even__succ__mod__exp,axiom,
    ! [Aa: 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)
     => ( 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),Aa)),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,Aa),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_4745_even__succ__mod__exp,axiom,
    ! [Aa: 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)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(Aa,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4746_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),one_one_nat)
    <=> ( Mb = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_4747_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Mb = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_4748_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_4749_dvd__add__triv__right__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),plus_p5714425477246183910nteger(B,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4750_dvd__add__triv__right__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(plus_plus_real(B),Aa))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4751_dvd__add__triv__right__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(B),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4752_dvd__add__triv__right__iff,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4753_dvd__add__triv__right__iff,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4754_dvd__add__triv__left__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),plus_p5714425477246183910nteger(Aa,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4755_dvd__add__triv__left__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(plus_plus_real(Aa),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4756_dvd__add__triv__left__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(Aa),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4757_dvd__add__triv__left__iff,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4758_dvd__add__triv__left__iff,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4759_div__dvd__div,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(B,Aa)),divide6298287555418463151nteger(C,Aa))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ) ).

% div_dvd_div
tff(fact_4760_div__dvd__div,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),Aa))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_4761_div__dvd__div,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Aa)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),Aa))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_4762_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_4763_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K),L) = L ).

% concat_bit_0
tff(fact_4764_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_4765_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_4766_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_4767_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_4768_dvd__times__right__cancel__iff,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4769_dvd__times__right__cancel__iff,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( ( Aa != 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4770_dvd__times__right__cancel__iff,axiom,
    ! [Aa: int,B: int,C: int] :
      ( ( Aa != 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4771_dvd__times__left__cancel__iff,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4772_dvd__times__left__cancel__iff,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( ( Aa != 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,Aa),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4773_dvd__times__left__cancel__iff,axiom,
    ! [Aa: int,B: int,C: int] :
      ( ( Aa != 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,Aa),B)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4774_dvd__mult__cancel__right,axiom,
    ! [Aa: 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,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4775_dvd__mult__cancel__right,axiom,
    ! [Aa: complex,C: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Aa),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4776_dvd__mult__cancel__right,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4777_dvd__mult__cancel__right,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4778_dvd__mult__cancel__right,axiom,
    ! [Aa: 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,Aa),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,Aa),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4779_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4780_dvd__mult__cancel__left,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),Aa)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Aa),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4781_dvd__mult__cancel__left,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(C),Aa)),aa_real_real(times_times_real(C),B))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4782_dvd__mult__cancel__left,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4783_dvd__mult__cancel__left,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4784_unit__prod,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),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,Aa),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_4785_unit__prod,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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,Aa),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_4786_unit__prod,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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,Aa),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_4787_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4788_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(plus_plus_real(aa_real_real(times_times_real(C),Aa)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4789_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C),Aa)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4790_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4791_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4792_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4793_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(plus_plus_real(B),aa_real_real(times_times_real(C),Aa)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4794_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(C),Aa)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4795_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4796_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4797_unit__div__1__div__1,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,Aa)) = Aa ) ) ).

% unit_div_1_div_1
tff(fact_4798_unit__div__1__div__1,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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),Aa)) = Aa ) ) ).

% unit_div_1_div_1
tff(fact_4799_unit__div__1__div__1,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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),Aa)) = Aa ) ) ).

% unit_div_1_div_1
tff(fact_4800_unit__div__1__unit,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,Aa)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_4801_unit__div__1__unit,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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),Aa)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_4802_unit__div__1__unit,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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),Aa)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_4803_unit__div,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),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,divide6298287555418463151nteger(Aa,B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_4804_unit__div,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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,Aa),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_4805_unit__div,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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,Aa),B)),one_one_int) ) ) ).

% unit_div
tff(fact_4806_dvd__mult__div__cancel,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),divide6298287555418463151nteger(B,Aa)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_4807_dvd__mult__div__cancel,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Aa)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_4808_dvd__mult__div__cancel,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Aa)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_4809_dvd__div__mult__self,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,Aa)),Aa) = B ) ) ).

% dvd_div_mult_self
tff(fact_4810_dvd__div__mult__self,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Aa)),Aa) = B ) ) ).

% dvd_div_mult_self
tff(fact_4811_dvd__div__mult__self,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Aa)),Aa) = B ) ) ).

% dvd_div_mult_self
tff(fact_4812_div__add,axiom,
    ! [C: code_integer,Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(Aa,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(Aa,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_add
tff(fact_4813_div__add,axiom,
    ! [C: nat,Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Aa)
     => ( 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,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_4814_div__add,axiom,
    ! [C: int,Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa)
     => ( 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,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_4815_div__diff,axiom,
    ! [C: code_integer,Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(minus_8373710615458151222nteger(Aa,B),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(Aa,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_diff
tff(fact_4816_div__diff,axiom,
    ! [C: int,Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa)
     => ( 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,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_diff
tff(fact_4817_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_4818_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_4819_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_4820_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_4821_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_4822_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_4823_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_4824_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_4825_unit__mult__div__div,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,Aa)) = divide6298287555418463151nteger(B,Aa) ) ) ).

% unit_mult_div_div
tff(fact_4826_unit__mult__div__div,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Aa) ) ) ).

% unit_mult_div_div
tff(fact_4827_unit__mult__div__div,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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),Aa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Aa) ) ) ).

% unit_mult_div_div
tff(fact_4828_unit__div__mult__self,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,Aa)),Aa) = B ) ) ).

% unit_div_mult_self
tff(fact_4829_unit__div__mult__self,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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),Aa)),Aa) = B ) ) ).

% unit_div_mult_self
tff(fact_4830_unit__div__mult__self,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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),Aa)),Aa) = B ) ) ).

% unit_div_mult_self
tff(fact_4831_pow__divides__pow__iff,axiom,
    ! [Nb: nat,Aa: 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,Aa),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,Aa),B) ) ) ).

% pow_divides_pow_iff
tff(fact_4832_pow__divides__pow__iff,axiom,
    ! [Nb: nat,Aa: 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,Aa),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,Aa),B) ) ) ).

% pow_divides_pow_iff
tff(fact_4833_even__mult__iff,axiom,
    ! [Aa: 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,Aa),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        | 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_4834_even__mult__iff,axiom,
    ! [Aa: 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,Aa),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        | 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_4835_even__mult__iff,axiom,
    ! [Aa: 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,Aa),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        | 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_4836_even__add,axiom,
    ! [Aa: 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))),plus_p5714425477246183910nteger(Aa,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
      <=> 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_4837_even__add,axiom,
    ! [Aa: 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,Aa),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
      <=> 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_4838_even__add,axiom,
    ! [Aa: 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,Aa),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
      <=> 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_4839_odd__add,axiom,
    ! [Aa: 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))),plus_p5714425477246183910nteger(Aa,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        <=> ~ 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_4840_odd__add,axiom,
    ! [Aa: 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,Aa),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        <=> ~ 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_4841_odd__add,axiom,
    ! [Aa: 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,Aa),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        <=> ~ 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_4842_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_4843_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_4844_even__mod__2__iff,axiom,
    ! [Aa: 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,Aa),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))),Aa) ) ).

% even_mod_2_iff
tff(fact_4845_even__mod__2__iff,axiom,
    ! [Aa: 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,Aa),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))),Aa) ) ).

% even_mod_2_iff
tff(fact_4846_even__mod__2__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),modulo364778990260209775nteger(Aa,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))),Aa) ) ).

% even_mod_2_iff
tff(fact_4847_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_4848_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_4849_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,Nb))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_4850_zero__le__power__eq__numeral,axiom,
    ! [Aa: 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,Aa),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),Aa) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4851_zero__le__power__eq__numeral,axiom,
    ! [Aa: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),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),Aa) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4852_zero__le__power__eq__numeral,axiom,
    ! [Aa: 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,Aa),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),Aa) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4853_power__less__zero__eq__numeral,axiom,
    ! [Aa: 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,Aa),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,Aa),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4854_power__less__zero__eq__numeral,axiom,
    ! [Aa: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),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,Aa),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4855_power__less__zero__eq__numeral,axiom,
    ! [Aa: 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,Aa),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,Aa),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4856_power__less__zero__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_4857_power__less__zero__eq,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),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,Aa),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_4858_power__less__zero__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_4859_even__plus__one__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(Aa,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))),Aa) ) ).

% even_plus_one_iff
tff(fact_4860_even__plus__one__iff,axiom,
    ! [Aa: 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,Aa),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))),Aa) ) ).

% even_plus_one_iff
tff(fact_4861_even__plus__one__iff,axiom,
    ! [Aa: 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),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))),Aa) ) ).

% even_plus_one_iff
tff(fact_4862_even__diff,axiom,
    ! [Aa: 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))),minus_8373710615458151222nteger(Aa,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(Aa,B)) ) ).

% even_diff
tff(fact_4863_even__diff,axiom,
    ! [Aa: 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,Aa),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,Aa),B)) ) ).

% even_diff
tff(fact_4864_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_4865_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_4866_zero__less__power__eq__numeral,axiom,
    ! [Aa: 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,Aa),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))
          & ( Aa != 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),Aa) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4867_zero__less__power__eq__numeral,axiom,
    ! [Aa: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),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))
          & ( Aa != 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),Aa) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4868_zero__less__power__eq__numeral,axiom,
    ! [Aa: 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,Aa),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))
          & ( Aa != 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),Aa) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4869_even__succ__div__2,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4870_even__succ__div__2,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( 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),Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4871_even__succ__div__2,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( 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)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4872_even__succ__div__two,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(Aa,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4873_even__succ__div__two,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4874_even__succ__div__two,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4875_odd__succ__div__two,axiom,
    ! [Aa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(Aa,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_4876_odd__succ__div__two,axiom,
    ! [Aa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_4877_odd__succ__div__two,axiom,
    ! [Aa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_4878_even__power,axiom,
    ! [Aa: 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(Aa),Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_4879_even__power,axiom,
    ! [Aa: 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,Aa),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_4880_even__power,axiom,
    ! [Aa: 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,Aa),Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_4881_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_4882_odd__two__times__div__two__succ,axiom,
    ! [Aa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) = Aa ) ) ).

% odd_two_times_div_two_succ
tff(fact_4883_odd__two__times__div__two__succ,axiom,
    ! [Aa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = Aa ) ) ).

% odd_two_times_div_two_succ
tff(fact_4884_odd__two__times__div__two__succ,axiom,
    ! [Aa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( 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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = Aa ) ) ).

% odd_two_times_div_two_succ
tff(fact_4885_power__le__zero__eq__numeral,axiom,
    ! [Aa: 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,Aa),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,Aa),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 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4886_power__le__zero__eq__numeral,axiom,
    ! [Aa: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),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,Aa),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 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4887_power__le__zero__eq__numeral,axiom,
    ! [Aa: 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,Aa),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,Aa),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 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4888_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))),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_4889_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_4890_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_4891_even__succ__div__exp,axiom,
    ! [Aa: 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)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = divide6298287555418463151nteger(Aa,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_4892_even__succ__div__exp,axiom,
    ! [Aa: 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)
     => ( 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),Aa)),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,Aa),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_4893_even__succ__div__exp,axiom,
    ! [Aa: 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)
     => ( 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),Aa)),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,Aa),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_4894_dvd__productE,axiom,
    ! [P2: nat,Aa: 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,Aa),B))
     => ~ ! [X2: nat,Y: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X2),Aa)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y),B) ) ) ) ).

% dvd_productE
tff(fact_4895_dvd__productE,axiom,
    ! [P2: int,Aa: 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,Aa),B))
     => ~ ! [X2: int,Y: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X2),Y) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X2),Aa)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),B) ) ) ) ).

% dvd_productE
tff(fact_4896_division__decomp,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B6: nat,C5: nat] :
          ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B6),C5) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B6),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C5),C) ) ) ).

% division_decomp
tff(fact_4897_division__decomp,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ? [B6: int,C5: int] :
          ( ( Aa = aa_int_int(aa_int_fun_int_int(times_times_int,B6),C5) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B6),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C5),C) ) ) ).

% division_decomp
tff(fact_4898_one__dvd,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),Aa) ).

% one_dvd
tff(fact_4899_one__dvd,axiom,
    ! [Aa: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,one_one_complex),Aa) ).

% one_dvd
tff(fact_4900_one__dvd,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),Aa) ).

% one_dvd
tff(fact_4901_one__dvd,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),Aa) ).

% one_dvd
tff(fact_4902_one__dvd,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),Aa) ).

% one_dvd
tff(fact_4903_one__dvd,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),Aa) ).

% one_dvd
tff(fact_4904_unit__imp__dvd,axiom,
    ! [B: code_integer,Aa: 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),Aa) ) ).

% unit_imp_dvd
tff(fact_4905_unit__imp__dvd,axiom,
    ! [B: nat,Aa: 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),Aa) ) ).

% unit_imp_dvd
tff(fact_4906_unit__imp__dvd,axiom,
    ! [B: int,Aa: 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),Aa) ) ).

% unit_imp_dvd
tff(fact_4907_dvd__unit__imp__unit,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),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,Aa),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_4908_dvd__unit__imp__unit,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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,Aa),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_4909_dvd__unit__imp__unit,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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,Aa),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_4910_dvdE,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa)
     => ~ ! [K3: code_integer] : Aa != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvdE
tff(fact_4911_dvdE,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),Aa)
     => ~ ! [K3: real] : Aa != aa_real_real(times_times_real(B),K3) ) ).

% dvdE
tff(fact_4912_dvdE,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),Aa)
     => ~ ! [K3: rat] : Aa != aa_rat_rat(times_times_rat(B),K3) ) ).

% dvdE
tff(fact_4913_dvdE,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Aa)
     => ~ ! [K3: nat] : Aa != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvdE
tff(fact_4914_dvdE,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Aa)
     => ~ ! [K3: int] : Aa != aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvdE
tff(fact_4915_dvdI,axiom,
    ! [Aa: code_integer,B: code_integer,K: code_integer] :
      ( ( Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa) ) ).

% dvdI
tff(fact_4916_dvdI,axiom,
    ! [Aa: real,B: real,K: real] :
      ( ( Aa = aa_real_real(times_times_real(B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),Aa) ) ).

% dvdI
tff(fact_4917_dvdI,axiom,
    ! [Aa: rat,B: rat,K: rat] :
      ( ( Aa = aa_rat_rat(times_times_rat(B),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),Aa) ) ).

% dvdI
tff(fact_4918_dvdI,axiom,
    ! [Aa: nat,B: nat,K: nat] :
      ( ( Aa = 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),Aa) ) ).

% dvdI
tff(fact_4919_dvdI,axiom,
    ! [Aa: int,B: int,K: int] :
      ( ( Aa = 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),Aa) ) ).

% dvdI
tff(fact_4920_dvd__def,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa)
    <=> ? [K2: code_integer] : Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).

% dvd_def
tff(fact_4921_dvd__def,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),Aa)
    <=> ? [K2: real] : Aa = aa_real_real(times_times_real(B),K2) ) ).

% dvd_def
tff(fact_4922_dvd__def,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),Aa)
    <=> ? [K2: rat] : Aa = aa_rat_rat(times_times_rat(B),K2) ) ).

% dvd_def
tff(fact_4923_dvd__def,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Aa)
    <=> ? [K2: nat] : Aa = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvd_def
tff(fact_4924_dvd__def,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Aa)
    <=> ? [K2: int] : Aa = aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvd_def
tff(fact_4925_dvd__mult,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_4926_dvd__mult,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(times_times_real(B),C)) ) ).

% dvd_mult
tff(fact_4927_dvd__mult,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult
tff(fact_4928_dvd__mult,axiom,
    ! [Aa: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_4929_dvd__mult,axiom,
    ! [Aa: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_4930_dvd__mult2,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_4931_dvd__mult2,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(times_times_real(B),C)) ) ).

% dvd_mult2
tff(fact_4932_dvd__mult2,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult2
tff(fact_4933_dvd__mult2,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_4934_dvd__mult2,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_4935_dvd__mult__left,axiom,
    ! [Aa: 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,Aa),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ).

% dvd_mult_left
tff(fact_4936_dvd__mult__left,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(Aa),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C) ) ).

% dvd_mult_left
tff(fact_4937_dvd__mult__left,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(Aa),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C) ) ).

% dvd_mult_left
tff(fact_4938_dvd__mult__left,axiom,
    ! [Aa: 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,Aa),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ).

% dvd_mult_left
tff(fact_4939_dvd__mult__left,axiom,
    ! [Aa: 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,Aa),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ).

% dvd_mult_left
tff(fact_4940_dvd__triv__left,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B)) ).

% dvd_triv_left
tff(fact_4941_dvd__triv__left,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(times_times_real(Aa),B)) ).

% dvd_triv_left
tff(fact_4942_dvd__triv__left,axiom,
    ! [Aa: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(times_times_rat(Aa),B)) ).

% dvd_triv_left
tff(fact_4943_dvd__triv__left,axiom,
    ! [Aa: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B)) ).

% dvd_triv_left
tff(fact_4944_dvd__triv__left,axiom,
    ! [Aa: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B)) ).

% dvd_triv_left
tff(fact_4945_mult__dvd__mono,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),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,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4946_mult__dvd__mono,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),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(times_times_real(Aa),C)),aa_real_real(times_times_real(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4947_mult__dvd__mono,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),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(Aa),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4948_mult__dvd__mono,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4949_mult__dvd__mono,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4950_dvd__mult__right,axiom,
    ! [Aa: 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,Aa),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_4951_dvd__mult__right,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(Aa),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_4952_dvd__mult__right,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(Aa),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_4953_dvd__mult__right,axiom,
    ! [Aa: 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,Aa),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_4954_dvd__mult__right,axiom,
    ! [Aa: 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,Aa),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_4955_dvd__triv__right,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Aa)) ).

% dvd_triv_right
tff(fact_4956_dvd__triv__right,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(times_times_real(B),Aa)) ).

% dvd_triv_right
tff(fact_4957_dvd__triv__right,axiom,
    ! [Aa: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(times_times_rat(B),Aa)) ).

% dvd_triv_right
tff(fact_4958_dvd__triv__right,axiom,
    ! [Aa: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Aa)) ).

% dvd_triv_right
tff(fact_4959_dvd__triv__right,axiom,
    ! [Aa: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,B),Aa)) ).

% dvd_triv_right
tff(fact_4960_dvd__add__right__iff,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_4961_dvd__add__right__iff,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_4962_dvd__add__right__iff,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_4963_dvd__add__right__iff,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_4964_dvd__add__right__iff,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_4965_dvd__add__left__iff,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B) ) ) ).

% dvd_add_left_iff
tff(fact_4966_dvd__add__left__iff,axiom,
    ! [Aa: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B) ) ) ).

% dvd_add_left_iff
tff(fact_4967_dvd__add__left__iff,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B) ) ) ).

% dvd_add_left_iff
tff(fact_4968_dvd__add__left__iff,axiom,
    ! [Aa: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B) ) ) ).

% dvd_add_left_iff
tff(fact_4969_dvd__add__left__iff,axiom,
    ! [Aa: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B) ) ) ).

% dvd_add_left_iff
tff(fact_4970_dvd__add,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),plus_p5714425477246183910nteger(B,C)) ) ) ).

% dvd_add
tff(fact_4971_dvd__add,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(plus_plus_real(B),C)) ) ) ).

% dvd_add
tff(fact_4972_dvd__add,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(B),C)) ) ) ).

% dvd_add
tff(fact_4973_dvd__add,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_4974_dvd__add,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).

% dvd_add
tff(fact_4975_dvd__diff__commute,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),minus_8373710615458151222nteger(C,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),minus_8373710615458151222nteger(B,C)) ) ).

% dvd_diff_commute
tff(fact_4976_dvd__diff__commute,axiom,
    ! [Aa: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% dvd_diff_commute
tff(fact_4977_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,Aa: 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),Aa)
       => ( divide6298287555418463151nteger(divide6298287555418463151nteger(Aa,D),divide6298287555418463151nteger(B,D)) = divide6298287555418463151nteger(Aa,B) ) ) ) ).

% div_div_div_same
tff(fact_4978_div__div__div__same,axiom,
    ! [D: nat,B: nat,Aa: 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),Aa)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),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,Aa),B) ) ) ) ).

% div_div_div_same
tff(fact_4979_div__div__div__same,axiom,
    ! [D: int,B: int,Aa: 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),Aa)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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,Aa),B) ) ) ) ).

% div_div_div_same
tff(fact_4980_dvd__div__eq__cancel,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger(Aa,C) = divide6298287555418463151nteger(B,C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
         => ( Aa = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4981_dvd__div__eq__cancel,axiom,
    ! [Aa: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
     => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),Aa)
       => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),B)
         => ( Aa = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4982_dvd__div__eq__cancel,axiom,
    ! [Aa: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(Aa),C) = aa_real_real(divide_divide_real(B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),Aa)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( Aa = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4983_dvd__div__eq__cancel,axiom,
    ! [Aa: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Aa),C) = aa_rat_rat(divide_divide_rat(B),C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( Aa = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4984_dvd__div__eq__cancel,axiom,
    ! [Aa: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),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),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( Aa = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4985_dvd__div__eq__cancel,axiom,
    ! [Aa: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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),Aa)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( Aa = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4986_dvd__div__eq__iff,axiom,
    ! [C: code_integer,Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( ( divide6298287555418463151nteger(Aa,C) = divide6298287555418463151nteger(B,C) )
        <=> ( Aa = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4987_dvd__div__eq__iff,axiom,
    ! [C: complex,Aa: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),Aa)
     => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),B)
       => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
        <=> ( Aa = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4988_dvd__div__eq__iff,axiom,
    ! [C: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),Aa)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(divide_divide_real(Aa),C) = aa_real_real(divide_divide_real(B),C) )
        <=> ( Aa = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4989_dvd__div__eq__iff,axiom,
    ! [C: rat,Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( aa_rat_rat(divide_divide_rat(Aa),C) = aa_rat_rat(divide_divide_rat(B),C) )
        <=> ( Aa = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4990_dvd__div__eq__iff,axiom,
    ! [C: nat,Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Aa)
     => ( 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),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( Aa = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4991_dvd__div__eq__iff,axiom,
    ! [C: int,Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa)
     => ( 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),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
        <=> ( Aa = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4992_dvd__power__same,axiom,
    ! [Xb: code_integer,Y3: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y3)
     => 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(Y3),Nb)) ) ).

% dvd_power_same
tff(fact_4993_dvd__power__same,axiom,
    ! [Xb: nat,Y3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y3)
     => 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,Y3),Nb)) ) ).

% dvd_power_same
tff(fact_4994_dvd__power__same,axiom,
    ! [Xb: real,Y3: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y3)
     => 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,Y3),Nb)) ) ).

% dvd_power_same
tff(fact_4995_dvd__power__same,axiom,
    ! [Xb: int,Y3: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y3)
     => 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,Y3),Nb)) ) ).

% dvd_power_same
tff(fact_4996_dvd__power__same,axiom,
    ! [Xb: complex,Y3: complex,Nb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y3)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),Nb)) ) ).

% dvd_power_same
tff(fact_4997_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,Aa: 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,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),C) ) ) ).

% mod_mod_cancel
tff(fact_4998_mod__mod__cancel,axiom,
    ! [C: int,B: int,Aa: 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,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) ) ) ).

% mod_mod_cancel
tff(fact_4999_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(Aa,B),C) = modulo364778990260209775nteger(Aa,C) ) ) ).

% mod_mod_cancel
tff(fact_5000_dvd__mod,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
     => ( 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,Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_5001_dvd__mod,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Mb)
     => ( 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,Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_5002_dvd__mod,axiom,
    ! [K: code_integer,Mb: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Mb)
     => ( 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),modulo364778990260209775nteger(Mb,Nb)) ) ) ).

% dvd_mod
tff(fact_5003_dvd__diff__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
     => ( 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,Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_5004_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb) ) ) ).

% dvd_pos_nat
tff(fact_5005_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_5006_bezout__add__nat,axiom,
    ! [Aa: nat,B: nat] :
    ? [D3: nat,X2: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),Aa)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D3) )
        | ( 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,Aa),Y)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_5007_bezout__lemma__nat,axiom,
    ! [D: nat,Aa: nat,B: nat,Xb: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),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,Aa),Y3)),D) ) )
         => ? [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Aa)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),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,Aa),B)),Y)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_5008_bezout1__nat,axiom,
    ! [Aa: nat,B: nat] :
    ? [D3: nat,X2: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),Aa)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)) = D3 )
        | ( 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,Aa),Y)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_5009_odd__numeral__BitM,axiom,
    ! [W: 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(bitM,W))) ).

% odd_numeral_BitM
tff(fact_5010_odd__numeral__BitM,axiom,
    ! [W: 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(bitM,W))) ).

% odd_numeral_BitM
tff(fact_5011_odd__numeral__BitM,axiom,
    ! [W: 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(bitM,W))) ).

% odd_numeral_BitM
tff(fact_5012_subset__divisors__dvd,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_gc,Aa))),collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_gc,B)))
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Aa),B) ) ).

% subset_divisors_dvd
tff(fact_5013_subset__divisors__dvd,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gd,Aa))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gd,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B) ) ).

% subset_divisors_dvd
tff(fact_5014_subset__divisors__dvd,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ge,Aa)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ge,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B) ) ).

% subset_divisors_dvd
tff(fact_5015_subset__divisors__dvd,axiom,
    ! [Aa: int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_gf,Aa))),collect_int(aa_int_fun_int_o(aTP_Lamm_gf,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B) ) ).

% subset_divisors_dvd
tff(fact_5016_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,Mb: 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(Mb),L),R2)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L)),R2) ).

% concat_bit_assoc
tff(fact_5017_strict__subset__divisors__dvd,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_set_complex_o(ord_less_set_complex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_gc,Aa))),collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_gc,B)))
    <=> ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Aa),B)
        & ~ aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),Aa) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5018_strict__subset__divisors__dvd,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gd,Aa))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gd,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Aa) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5019_strict__subset__divisors__dvd,axiom,
    ! [Aa: int,B: int] :
      ( aa_set_int_o(ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_gf,Aa))),collect_int(aa_int_fun_int_o(aTP_Lamm_gf,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Aa) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5020_strict__subset__divisors__dvd,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ge,Aa)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ge,B)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5021_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_gf,I))) ) ).

% finite_divisors_int
tff(fact_5022_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_5023_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_5024_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_5025_pinf_I9_J,axiom,
    ! [D: code_integer,S: 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),plus_p5714425477246183910nteger(X3,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)) ) ) ).

% pinf(9)
tff(fact_5026_pinf_I9_J,axiom,
    ! [D: real,S: 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(plus_plus_real(X3),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X3),S)) ) ) ).

% pinf(9)
tff(fact_5027_pinf_I9_J,axiom,
    ! [D: rat,S: 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),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S)) ) ) ).

% pinf(9)
tff(fact_5028_pinf_I9_J,axiom,
    ! [D: nat,S: 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),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% pinf(9)
tff(fact_5029_pinf_I9_J,axiom,
    ! [D: int,S: 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),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% pinf(9)
tff(fact_5030_pinf_I10_J,axiom,
    ! [D: code_integer,S: 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),plus_p5714425477246183910nteger(X3,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)) ) ) ).

% pinf(10)
tff(fact_5031_pinf_I10_J,axiom,
    ! [D: real,S: 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(plus_plus_real(X3),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X3),S)) ) ) ).

% pinf(10)
tff(fact_5032_pinf_I10_J,axiom,
    ! [D: rat,S: 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),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S)) ) ) ).

% pinf(10)
tff(fact_5033_pinf_I10_J,axiom,
    ! [D: nat,S: 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),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% pinf(10)
tff(fact_5034_pinf_I10_J,axiom,
    ! [D: int,S: 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),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% pinf(10)
tff(fact_5035_minf_I9_J,axiom,
    ! [D: code_integer,S: 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),plus_p5714425477246183910nteger(X3,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)) ) ) ).

% minf(9)
tff(fact_5036_minf_I9_J,axiom,
    ! [D: real,S: 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(plus_plus_real(X3),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X3),S)) ) ) ).

% minf(9)
tff(fact_5037_minf_I9_J,axiom,
    ! [D: rat,S: 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),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S)) ) ) ).

% minf(9)
tff(fact_5038_minf_I9_J,axiom,
    ! [D: nat,S: 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),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% minf(9)
tff(fact_5039_minf_I9_J,axiom,
    ! [D: int,S: 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),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% minf(9)
tff(fact_5040_minf_I10_J,axiom,
    ! [D: code_integer,S: 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),plus_p5714425477246183910nteger(X3,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)) ) ) ).

% minf(10)
tff(fact_5041_minf_I10_J,axiom,
    ! [D: real,S: 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(plus_plus_real(X3),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X3),S)) ) ) ).

% minf(10)
tff(fact_5042_minf_I10_J,axiom,
    ! [D: rat,S: 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),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S)) ) ) ).

% minf(10)
tff(fact_5043_minf_I10_J,axiom,
    ! [D: nat,S: 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),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% minf(10)
tff(fact_5044_minf_I10_J,axiom,
    ! [D: int,S: 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),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% minf(10)
tff(fact_5045_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa)
     => ( ( divide6298287555418463151nteger(Aa,B) = zero_z3403309356797280102nteger )
      <=> ( Aa = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5046_dvd__div__eq__0__iff,axiom,
    ! [B: complex,Aa: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),Aa)
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) = zero_zero_complex )
      <=> ( Aa = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5047_dvd__div__eq__0__iff,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),Aa)
     => ( ( aa_real_real(divide_divide_real(Aa),B) = zero_zero_real )
      <=> ( Aa = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5048_dvd__div__eq__0__iff,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),Aa)
     => ( ( aa_rat_rat(divide_divide_rat(Aa),B) = zero_zero_rat )
      <=> ( Aa = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5049_dvd__div__eq__0__iff,axiom,
    ! [B: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Aa)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B) = zero_zero_nat )
      <=> ( Aa = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5050_dvd__div__eq__0__iff,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Aa)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B) = zero_zero_int )
      <=> ( Aa = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5051_is__unit__mult__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),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_5052_is__unit__mult__iff,axiom,
    ! [Aa: 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,Aa),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_5053_is__unit__mult__iff,axiom,
    ! [Aa: 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,Aa),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_5054_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,Aa: 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),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_5055_dvd__mult__unit__iff,axiom,
    ! [B: nat,Aa: 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),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_5056_dvd__mult__unit__iff,axiom,
    ! [B: int,Aa: 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),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_5057_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,Aa: 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,Aa),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_5058_mult__unit__dvd__iff,axiom,
    ! [B: nat,Aa: 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,Aa),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_5059_mult__unit__dvd__iff,axiom,
    ! [B: int,Aa: 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,Aa),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_5060_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,Aa: 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),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5061_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,Aa: 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),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5062_dvd__mult__unit__iff_H,axiom,
    ! [B: int,Aa: 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),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5063_mult__unit__dvd__iff_H,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5064_mult__unit__dvd__iff_H,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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,Aa),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5065_mult__unit__dvd__iff_H,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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,Aa),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5066_unit__mult__left__cancel,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_5067_unit__mult__left__cancel,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_5068_unit__mult__left__cancel,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_5069_unit__mult__right__cancel,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_5070_unit__mult__right__cancel,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_5071_unit__mult__right__cancel,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),Aa) = aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_5072_dvd__div__unit__iff,axiom,
    ! [B: code_integer,Aa: 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),divide6298287555418463151nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5073_dvd__div__unit__iff,axiom,
    ! [B: nat,Aa: 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),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5074_dvd__div__unit__iff,axiom,
    ! [B: int,Aa: 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),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5075_div__unit__dvd__iff,axiom,
    ! [B: code_integer,Aa: 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,divide6298287555418463151nteger(Aa,B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5076_div__unit__dvd__iff,axiom,
    ! [B: nat,Aa: 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,Aa),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5077_div__unit__dvd__iff,axiom,
    ! [B: int,Aa: 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,Aa),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5078_unit__div__cancel,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(B,Aa) = divide6298287555418463151nteger(C,Aa) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_5079_unit__div__cancel,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),Aa) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_5080_unit__div__cancel,axiom,
    ! [Aa: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Aa) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),Aa) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_5081_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,Aa: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(Aa,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5082_div__mult__div__if__dvd,axiom,
    ! [B: nat,Aa: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Aa)
     => ( 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,Aa),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5083_div__mult__div__if__dvd,axiom,
    ! [B: int,Aa: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Aa)
     => ( 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,Aa),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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5084_dvd__mult__imp__div,axiom,
    ! [Aa: 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,Aa),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),divide6298287555418463151nteger(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_5085_dvd__mult__imp__div,axiom,
    ! [Aa: 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,Aa),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_5086_dvd__mult__imp__div,axiom,
    ! [Aa: 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,Aa),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_5087_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),Aa)
     => ( divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(Aa,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5088_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,Aa: 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)),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),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,Aa),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5089_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,Aa: 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)),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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,Aa),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5090_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,Aa: 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),Aa)
       => ( divide6298287555418463151nteger(Aa,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(Aa,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_5091_div__div__eq__right,axiom,
    ! [C: nat,B: nat,Aa: 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),Aa)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),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,Aa),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_5092_div__div__eq__right,axiom,
    ! [C: int,B: int,Aa: 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),Aa)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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,Aa),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_5093_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B),C) ) ) ).

% div_mult_swap
tff(fact_5094_div__mult__swap,axiom,
    ! [C: nat,B: nat,Aa: 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),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,Aa),B)),C) ) ) ).

% div_mult_swap
tff(fact_5095_div__mult__swap,axiom,
    ! [C: int,B: int,Aa: 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),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,Aa),B)),C) ) ) ).

% div_mult_swap
tff(fact_5096_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),Aa) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Aa),C) ) ) ).

% dvd_div_mult
tff(fact_5097_dvd__div__mult,axiom,
    ! [C: nat,B: nat,Aa: 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)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Aa)),C) ) ) ).

% dvd_div_mult
tff(fact_5098_dvd__div__mult,axiom,
    ! [C: int,B: int,Aa: 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)),Aa) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Aa)),C) ) ) ).

% dvd_div_mult
tff(fact_5099_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(Aa,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(Aa,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5100_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,Aa: 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,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5101_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,Aa: 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,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5102_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(Aa,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(Aa,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5103_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5104_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5105_div__power,axiom,
    ! [B: code_integer,Aa: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(Aa,B)),Nb) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% div_power
tff(fact_5106_div__power,axiom,
    ! [B: nat,Aa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% div_power
tff(fact_5107_div__power,axiom,
    ! [B: int,Aa: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Aa)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)),Nb) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% div_power
tff(fact_5108_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,Aa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5109_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5110_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5111_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5112_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,Aa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5113_power__le__dvd,axiom,
    ! [Aa: code_integer,Nb: nat,B: code_integer,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5114_power__le__dvd,axiom,
    ! [Aa: nat,Nb: nat,B: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5115_power__le__dvd,axiom,
    ! [Aa: real,Nb: nat,B: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5116_power__le__dvd,axiom,
    ! [Aa: int,Nb: nat,B: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5117_power__le__dvd,axiom,
    ! [Aa: complex,Nb: nat,B: complex,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5118_dvd__power__le,axiom,
    ! [Xb: code_integer,Y3: code_integer,Nb: nat,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),Mb)) ) ) ).

% dvd_power_le
tff(fact_5119_dvd__power__le,axiom,
    ! [Xb: nat,Y3: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),Mb)) ) ) ).

% dvd_power_le
tff(fact_5120_dvd__power__le,axiom,
    ! [Xb: real,Y3: real,Nb: nat,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),Mb)) ) ) ).

% dvd_power_le
tff(fact_5121_dvd__power__le,axiom,
    ! [Xb: int,Y3: int,Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),Mb)) ) ) ).

% dvd_power_le
tff(fact_5122_dvd__power__le,axiom,
    ! [Xb: complex,Y3: complex,Nb: nat,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),Mb)) ) ) ).

% dvd_power_le
tff(fact_5123_mod__eq__dvd__iff,axiom,
    ! [Aa: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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,Aa),B)) ) ).

% mod_eq_dvd_iff
tff(fact_5124_mod__eq__dvd__iff,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(Aa,C) = modulo364778990260209775nteger(B,C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(Aa,B)) ) ).

% mod_eq_dvd_iff
tff(fact_5125_bezout__add__strong__nat,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa != zero_zero_nat )
     => ? [D3: nat,X2: nat,Y: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),Aa)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_5126_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_5127_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_5128_dvd__diffD,axiom,
    ! [K: nat,Mb: 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,Mb),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),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb) ) ) ) ).

% dvd_diffD
tff(fact_5129_dvd__diffD1,axiom,
    ! [K: nat,Mb: 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,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_5130_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_5131_zdvd__antisym__nonneg,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb)
           => ( Mb = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_5132_zdvd__mono,axiom,
    ! [K: int,Mb: int,Ta: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Ta)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Ta)) ) ) ).

% zdvd_mono
tff(fact_5133_zdvd__mult__cancel,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),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,Mb),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_5134_dbl__def,axiom,
    ! [Xb: real] : aa_real_real(neg_numeral_dbl_real,Xb) = aa_real_real(plus_plus_real(Xb),Xb) ).

% dbl_def
tff(fact_5135_dbl__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_numeral_dbl_rat,Xb) = aa_rat_rat(plus_plus_rat(Xb),Xb) ).

% dbl_def
tff(fact_5136_dbl__def,axiom,
    ! [Xb: int] : aa_int_int(neg_numeral_dbl_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb) ).

% dbl_def
tff(fact_5137_zdvd__period,axiom,
    ! [Aa: int,D: int,Xb: int,Ta: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Ta))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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_5138_zdvd__reduce,axiom,
    ! [K: int,Nb: int,Mb: 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),Mb)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb) ) ).

% zdvd_reduce
tff(fact_5139_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_5140_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_5141_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gd,Mb))) ) ).

% finite_divisors_nat
tff(fact_5142_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y3: 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,Y3),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))),Y3) )
       => ( Xb = Y3 ) ) ) ).

% div2_even_ext_nat
tff(fact_5143_unit__dvdE,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ~ ( ( Aa != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C3) ) ) ).

% unit_dvdE
tff(fact_5144_unit__dvdE,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ~ ( ( Aa != zero_zero_nat )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C3) ) ) ).

% unit_dvdE
tff(fact_5145_unit__dvdE,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ~ ( ( Aa != zero_zero_int )
         => ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C3) ) ) ).

% unit_dvdE
tff(fact_5146_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),plus_p5714425477246183910nteger(X,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5147_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X: complex] : aa_complex_o(P,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,L),X))
    <=> ? [X: complex] :
          ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,L),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X),zero_zero_complex))
          & aa_complex_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5148_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X: real] : aa_real_o(P,aa_real_real(times_times_real(L),X))
    <=> ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(plus_plus_real(X),zero_zero_real))
          & aa_real_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5149_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_5150_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_5151_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_5152_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_5153_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_5154_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_5155_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(Aa,B) = zero_z3403309356797280102nteger )
      <=> ( Aa = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5156_unit__div__eq__0__iff,axiom,
    ! [B: nat,Aa: 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,Aa),B) = zero_zero_nat )
      <=> ( Aa = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5157_unit__div__eq__0__iff,axiom,
    ! [B: int,Aa: 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,Aa),B) = zero_zero_int )
      <=> ( Aa = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5158_dvd__div__div__eq__mult,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( divide6298287555418463151nteger(B,Aa) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5159_dvd__div__div__eq__mult,axiom,
    ! [Aa: nat,C: nat,B: nat,D: nat] :
      ( ( Aa != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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),Aa) = 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,Aa),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5160_dvd__div__div__eq__mult,axiom,
    ! [Aa: int,C: int,B: int,D: int] :
      ( ( Aa != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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),Aa) = 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,Aa),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5161_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,Aa: 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,Aa),divide6298287555418463151nteger(B,C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5162_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,Aa: 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,Aa),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,Aa),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5163_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,Aa: 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,Aa),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,Aa),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5164_div__dvd__iff__mult,axiom,
    ! [B: code_integer,Aa: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(Aa,B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5165_div__dvd__iff__mult,axiom,
    ! [B: nat,Aa: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5166_div__dvd__iff__mult,axiom,
    ! [B: int,Aa: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Aa)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5167_dvd__div__eq__mult,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),B)
       => ( ( divide6298287555418463151nteger(B,Aa) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5168_dvd__div__eq__mult,axiom,
    ! [Aa: nat,B: nat,C: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Aa) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5169_dvd__div__eq__mult,axiom,
    ! [Aa: int,B: int,C: int] :
      ( ( Aa != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Aa) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5170_unit__eq__div1,axiom,
    ! [B: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(Aa,B) = C )
      <=> ( Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_5171_unit__eq__div1,axiom,
    ! [B: nat,Aa: 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,Aa),B) = C )
      <=> ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_5172_unit__eq__div1,axiom,
    ! [B: int,Aa: 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,Aa),B) = C )
      <=> ( Aa = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_5173_unit__eq__div2,axiom,
    ! [B: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( Aa = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_5174_unit__eq__div2,axiom,
    ! [B: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_5175_unit__eq__div2,axiom,
    ! [B: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( Aa = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_5176_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,Aa: 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),Aa)
       => ( divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(Aa,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_5177_div__mult__unit2,axiom,
    ! [C: nat,B: nat,Aa: 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),Aa)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),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,Aa),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_5178_div__mult__unit2,axiom,
    ! [C: int,B: int,Aa: 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),Aa)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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,Aa),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_5179_unit__div__commute,axiom,
    ! [B: code_integer,Aa: 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,divide6298287555418463151nteger(Aa,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C),B) ) ) ).

% unit_div_commute
tff(fact_5180_unit__div__commute,axiom,
    ! [B: nat,Aa: 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,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),B) ) ) ).

% unit_div_commute
tff(fact_5181_unit__div__commute,axiom,
    ! [B: int,Aa: 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,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),B) ) ) ).

% unit_div_commute
tff(fact_5182_unit__div__mult__swap,axiom,
    ! [C: code_integer,Aa: 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,Aa),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_5183_unit__div__mult__swap,axiom,
    ! [C: nat,Aa: 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,Aa),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,Aa),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_5184_unit__div__mult__swap,axiom,
    ! [C: int,Aa: 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,Aa),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,Aa),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_5185_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,Aa: 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)
       => ( divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(Aa,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5186_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,Aa: 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,Aa),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,Aa),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5187_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,Aa: 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,Aa),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,Aa),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5188_inf__period_I4_J,axiom,
    ! [D: code_integer,D2: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D2)
     => ! [X3: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D2)),Ta)) ) ) ).

% inf_period(4)
tff(fact_5189_inf__period_I4_J,axiom,
    ! [D: real,D2: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D2)
     => ! [X3: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X3),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K4),D2))),Ta)) ) ) ).

% inf_period(4)
tff(fact_5190_inf__period_I4_J,axiom,
    ! [D: rat,D2: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D2)
     => ! [X3: rat,K4: 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(aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D2))),Ta)) ) ) ).

% inf_period(4)
tff(fact_5191_inf__period_I4_J,axiom,
    ! [D: int,D2: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X3: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,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,K4),D2))),Ta)) ) ) ).

% inf_period(4)
tff(fact_5192_inf__period_I3_J,axiom,
    ! [D: code_integer,D2: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D2)
     => ! [X3: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D2)),Ta)) ) ) ).

% inf_period(3)
tff(fact_5193_inf__period_I3_J,axiom,
    ! [D: real,D2: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D2)
     => ! [X3: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X3),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K4),D2))),Ta)) ) ) ).

% inf_period(3)
tff(fact_5194_inf__period_I3_J,axiom,
    ! [D: rat,D2: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D2)
     => ! [X3: rat,K4: 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(aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D2))),Ta)) ) ) ).

% inf_period(3)
tff(fact_5195_inf__period_I3_J,axiom,
    ! [D: int,D2: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X3: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,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,K4),D2))),Ta)) ) ) ).

% inf_period(3)
tff(fact_5196_is__unit__power__iff,axiom,
    ! [Aa: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5197_is__unit__power__iff,axiom,
    ! [Aa: 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,Aa),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5198_is__unit__power__iff,axiom,
    ! [Aa: 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,Aa),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5199_unit__imp__mod__eq__0,axiom,
    ! [B: nat,Aa: 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,Aa),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_5200_unit__imp__mod__eq__0,axiom,
    ! [B: int,Aa: 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,Aa),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_5201_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(Aa,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_5202_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_5203_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_5204_dvd__mult__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_5205_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_5206_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_5207_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),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,Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_5208_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_5209_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_5210_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_5211_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_5212_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_5213_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_5214_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_5215_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_5216_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_5217_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_5218_evenE,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ~ ! [B3: code_integer] : Aa != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_5219_evenE,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ~ ! [B3: nat] : Aa != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_5220_evenE,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ~ ! [B3: int] : Aa != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_5221_odd__even__add,axiom,
    ! [Aa: 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)
     => ( ~ 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))),plus_p5714425477246183910nteger(Aa,B)) ) ) ).

% odd_even_add
tff(fact_5222_odd__even__add,axiom,
    ! [Aa: 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)
     => ( ~ 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,Aa),B)) ) ) ).

% odd_even_add
tff(fact_5223_odd__even__add,axiom,
    ! [Aa: 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)
     => ( ~ 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,Aa),B)) ) ) ).

% odd_even_add
tff(fact_5224_is__unitE,axiom,
    ! [Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ~ ( ( Aa != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B3),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,Aa) = B3 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B3) = Aa )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B3) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,Aa) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5225_is__unitE,axiom,
    ! [Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ~ ( ( Aa != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B3),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),Aa) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = Aa )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),Aa) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5226_is__unitE,axiom,
    ! [Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ~ ( ( Aa != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B3),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),Aa) = B3 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B3) = Aa )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B3) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),Aa) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5227_is__unit__div__mult__cancel__left,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5228_is__unit__div__mult__cancel__left,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa != 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5229_is__unit__div__mult__cancel__left,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa != 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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5230_is__unit__div__mult__cancel__right,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),Aa)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5231_is__unit__div__mult__cancel__right,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa != 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5232_is__unit__div__mult__cancel__right,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa != 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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,B),Aa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5233_bit__eq__rec,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( Aa = B )
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) )
        & ( divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(B,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_5234_bit__eq__rec,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa = B )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        <=> 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,Aa),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_5235_bit__eq__rec,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa = B )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        <=> 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,Aa),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_5236_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_5237_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_5238_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_5239_dvd__power__iff,axiom,
    ! [Xb: code_integer,Mb: nat,Nb: nat] :
      ( ( Xb != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_5240_dvd__power__iff,axiom,
    ! [Xb: nat,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_5241_dvd__power__iff,axiom,
    ! [Xb: int,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Mb)),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,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_5242_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_5243_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_5244_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_5245_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_5246_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_5247_dvd__power,axiom,
    ! [Nb: nat,Xb: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) ) ).

% dvd_power
tff(fact_5248_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_5249_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_5250_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_5251_power__dvd__imp__le,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_5252_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,Mb: 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),Mb))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_5253_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_5254_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_5255_bset_I9_J,axiom,
    ! [D: int,D2: int,B2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B2)
                 => ( 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),D2)),Ta)) ) ) ) ).

% bset(9)
tff(fact_5256_bset_I10_J,axiom,
    ! [D: int,D2: int,B2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B2)
                 => ( 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),D2)),Ta)) ) ) ) ).

% bset(10)
tff(fact_5257_aset_I9_J,axiom,
    ! [D: int,D2: int,A: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A)
                 => ( 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),D2)),Ta)) ) ) ) ).

% aset(9)
tff(fact_5258_aset_I10_J,axiom,
    ! [D: int,D2: int,A: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A)
                 => ( 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),D2)),Ta)) ) ) ) ).

% aset(10)
tff(fact_5259_numeral__BitM,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bitM,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,numera6690914467698888265omplex(aa_num_num(bit0,Nb))),one_one_complex) ).

% numeral_BitM
tff(fact_5260_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bitM,Nb)) = aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb))),one_one_real) ).

% numeral_BitM
tff(fact_5261_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,Nb)) = aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb))),one_one_rat) ).

% numeral_BitM
tff(fact_5262_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bitM,Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),one_one_int) ).

% numeral_BitM
tff(fact_5263_even__two__times__div__two,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = Aa ) ) ).

% even_two_times_div_two
tff(fact_5264_even__two__times__div__two,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( 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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = Aa ) ) ).

% even_two_times_div_two
tff(fact_5265_even__two__times__div__two,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( 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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = Aa ) ) ).

% even_two_times_div_two
tff(fact_5266_even__iff__mod__2__eq__zero,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5267_even__iff__mod__2__eq__zero,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5268_even__iff__mod__2__eq__zero,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
    <=> ( modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5269_odd__iff__mod__2__eq__one,axiom,
    ! [Aa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5270_odd__iff__mod__2__eq__one,axiom,
    ! [Aa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5271_odd__iff__mod__2__eq__one,axiom,
    ! [Aa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
    <=> ( modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5272_power__mono__odd,axiom,
    ! [Nb: nat,Aa: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_5273_power__mono__odd,axiom,
    ! [Nb: nat,Aa: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_5274_power__mono__odd,axiom,
    ! [Nb: nat,Aa: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_5275_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_5276_dvd__power__iff__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),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,Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_5277_even__unset__bit__iff,axiom,
    ! [Mb: nat,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Mb,Aa))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5278_even__unset__bit__iff,axiom,
    ! [Mb: nat,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),Aa))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5279_even__unset__bit__iff,axiom,
    ! [Mb: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),Aa))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5280_even__set__bit__iff,axiom,
    ! [Mb: nat,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Mb,Aa))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5281_even__set__bit__iff,axiom,
    ! [Mb: nat,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Mb),Aa))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5282_even__set__bit__iff,axiom,
    ! [Mb: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),Aa))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5283_even__flip__bit__iff,axiom,
    ! [Mb: nat,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Mb,Aa))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5284_even__flip__bit__iff,axiom,
    ! [Mb: nat,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Mb),Aa))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5285_even__flip__bit__iff,axiom,
    ! [Mb: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),Aa))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5286_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_5287_oddE,axiom,
    ! [Aa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ~ ! [B3: code_integer] : Aa != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B3),one_one_Code_integer) ) ).

% oddE
tff(fact_5288_oddE,axiom,
    ! [Aa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ~ ! [B3: nat] : Aa != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_5289_oddE,axiom,
    ! [Aa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ~ ! [B3: int] : Aa != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_5290_parity__cases,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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))),Aa)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_5291_parity__cases,axiom,
    ! [Aa: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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))),Aa)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_5292_parity__cases,axiom,
    ! [Aa: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
       => ( modulo364778990260209775nteger(Aa,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))),Aa)
         => ( modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_5293_mod2__eq__if,axiom,
    ! [Aa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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))),Aa),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_5294_mod2__eq__if,axiom,
    ! [Aa: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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))),Aa),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_5295_mod2__eq__if,axiom,
    ! [Aa: code_integer] :
      modulo364778990260209775nteger(Aa,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))),Aa),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_5296_zero__le__even__power,axiom,
    ! [Nb: nat,Aa: 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,Aa),Nb)) ) ).

% zero_le_even_power
tff(fact_5297_zero__le__even__power,axiom,
    ! [Nb: nat,Aa: 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(Aa),Nb)) ) ).

% zero_le_even_power
tff(fact_5298_zero__le__even__power,axiom,
    ! [Nb: nat,Aa: 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,Aa),Nb)) ) ).

% zero_le_even_power
tff(fact_5299_zero__le__odd__power,axiom,
    ! [Nb: nat,Aa: 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,Aa),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ) ).

% zero_le_odd_power
tff(fact_5300_zero__le__odd__power,axiom,
    ! [Nb: nat,Aa: 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(Aa),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ) ).

% zero_le_odd_power
tff(fact_5301_zero__le__odd__power,axiom,
    ! [Nb: nat,Aa: 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,Aa),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ) ).

% zero_le_odd_power
tff(fact_5302_zero__le__power__eq,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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),Aa) ) ) ) ).

% zero_le_power_eq
tff(fact_5303_zero__le__power__eq,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),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),Aa) ) ) ) ).

% zero_le_power_eq
tff(fact_5304_zero__le__power__eq,axiom,
    ! [Aa: 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,Aa),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),Aa) ) ) ) ).

% zero_le_power_eq
tff(fact_5305_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N: nat] : Xb != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_5306_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Aa: nat,B: nat] :
      ( ! [A2: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A2),B3)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B3),A2) )
     => ( ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A2),zero_zero_nat)
       => ( ! [A2: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A2),B3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Aa),B) ) ) ) ).

% Euclid_induct
tff(fact_5307_zero__less__power__eq,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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)
          & ( Aa != 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),Aa) ) ) ) ).

% zero_less_power_eq
tff(fact_5308_zero__less__power__eq,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),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)
          & ( Aa != 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),Aa) ) ) ) ).

% zero_less_power_eq
tff(fact_5309_zero__less__power__eq,axiom,
    ! [Aa: 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,Aa),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)
          & ( Aa != 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),Aa) ) ) ) ).

% zero_less_power_eq
tff(fact_5310_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),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,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_5311_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_5312_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),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,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_5313_power__le__zero__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),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 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5314_power__le__zero__eq,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),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,Aa),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 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5315_power__le__zero__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),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 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5316_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_5317_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),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,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5318_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5319_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),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,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5320_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_5321_even__mult__exp__div__exp__iff,axiom,
    ! [Aa: code_integer,Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)),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),Mb)
        | ( 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,Mb),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(Aa,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),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5322_even__mult__exp__div__exp__iff,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5323_even__mult__exp__div__exp__iff,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),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),Mb)
        | ( 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,Mb),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5324_set__encode__insert,axiom,
    ! [A: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(Nb),A)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A)) = 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,A)) ) ) ) ).

% set_encode_insert
tff(fact_5325_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_5326_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y3: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y3 )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y3 != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y3 != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y3 != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_5327_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gg),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5328_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gh),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5329_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_gi,unique3479559517661332726nteger(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5330_flip__bit__0,axiom,
    ! [Aa: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,Aa) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_5331_flip__bit__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,zero_zero_nat),Aa) = 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))),Aa))),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_5332_flip__bit__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),Aa) = 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))),Aa))),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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_5333_signed__take__bit__Suc,axiom,
    ! [Nb: nat,Aa: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Nb),Aa) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,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))),bit_ri6519982836138164636nteger(Nb,divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5334_signed__take__bit__Suc,axiom,
    ! [Nb: nat,Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5335_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat,X23: product_prod_nat_nat] : size_o8335143837870341156at_nat(Xb,some_P7363390416028606310at_nat(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(Xb,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5336_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_nat_nat,X23: nat] : size_option_nat(Xb,some_nat(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Xb,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5337_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_num_nat,X23: num] : size_option_num(Xb,some_num(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(Xb,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5338_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_5339_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_5340_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_5341_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_5342_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_5343_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_5344_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5345_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5346_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5347_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5348_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5349_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5350_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5351_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5352_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5353_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = one_one_int )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5354_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = one_one_Code_integer )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5355_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex($true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_5356_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real($true) = one_one_real ).

% of_bool_eq(2)
tff(fact_5357_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_5358_of__bool__eq_I2_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_5359_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_5360_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger($true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_5361_signed__take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_5362_replicate__eq__replicate,axiom,
    ! [Mb: nat,Xb: vEBT_VEBT,Nb: nat,Y3: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(Mb,Xb) = replicate_VEBT_VEBT(Nb,Y3) )
    <=> ( ( Mb = Nb )
        & ( ( Mb != zero_zero_nat )
         => ( Xb = Y3 ) ) ) ) ).

% replicate_eq_replicate
tff(fact_5363_length__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_5364_length__replicate,axiom,
    ! [Nb: nat,Xb: $o] : size_size_list_o(replicate_o(Nb,(Xb))) = Nb ).

% length_replicate
tff(fact_5365_length__replicate,axiom,
    ! [Nb: nat,Xb: int] : size_size_list_int(replicate_int(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_5366_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        | (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_or_iff
tff(fact_5367_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        | (Q) )) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_or_iff
tff(fact_5368_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        | (Q) )) = ord_max_Code_integer(zero_n356916108424825756nteger((P)),zero_n356916108424825756nteger((Q))) ).

% of_bool_or_iff
tff(fact_5369_case__prod__conv,axiom,
    ! [F: fun_na4138050610385705875at_nat,Aa: nat,B: nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Aa),B)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,Aa),B) ).

% case_prod_conv
tff(fact_5370_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_o,Aa: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Aa),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(F,Aa),B) ) ).

% case_prod_conv
tff(fact_5371_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,Aa: int,B: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Aa),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,Aa),B) ).

% case_prod_conv
tff(fact_5372_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,Aa: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Aa),B))
    <=> aa_int_o(aa_int_fun_int_o(F,Aa),B) ) ).

% case_prod_conv
tff(fact_5373_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,Aa: int,B: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(F,Aa),B) ).

% case_prod_conv
tff(fact_5374_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5375_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5376_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5377_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5378_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5379_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5380_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5381_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5382_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5383_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5384_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n1201886186963655149omplex(~ (P)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),zero_n1201886186963655149omplex((P))) ).

% of_bool_not_iff
tff(fact_5385_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n3304061248610475627l_real(~ (P)) = aa_real_real(minus_minus_real(one_one_real),zero_n3304061248610475627l_real((P))) ).

% of_bool_not_iff
tff(fact_5386_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat((P))) ).

% of_bool_not_iff
tff(fact_5387_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% of_bool_not_iff
tff(fact_5388_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n356916108424825756nteger(~ (P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger((P))) ).

% of_bool_not_iff
tff(fact_5389_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_5390_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_5391_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_5392_in__set__replicate,axiom,
    ! [Xb: complex,Nb: nat,Y3: complex] :
      ( aa_set_complex_o(member_complex(Xb),set_complex2(replicate_complex(Nb,Y3)))
    <=> ( ( Xb = Y3 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5393_in__set__replicate,axiom,
    ! [Xb: real,Nb: nat,Y3: real] :
      ( aa_set_real_o(member_real(Xb),set_real2(replicate_real(Nb,Y3)))
    <=> ( ( Xb = Y3 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5394_in__set__replicate,axiom,
    ! [Xb: set_nat,Nb: nat,Y3: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xb),set_set_nat2(replicate_set_nat(Nb,Y3)))
    <=> ( ( Xb = Y3 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5395_in__set__replicate,axiom,
    ! [Xb: nat,Nb: nat,Y3: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(replicate_nat(Nb,Y3)))
    <=> ( ( Xb = Y3 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5396_in__set__replicate,axiom,
    ! [Xb: int,Nb: nat,Y3: int] :
      ( aa_set_int_o(member_int(Xb),set_int2(replicate_int(Nb,Y3)))
    <=> ( ( Xb = Y3 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5397_in__set__replicate,axiom,
    ! [Xb: vEBT_VEBT,Nb: nat,Y3: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y3)))
    <=> ( ( Xb = Y3 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5398_Bex__set__replicate,axiom,
    ! [Nb: nat,Aa: int,P: fun_int_o] :
      ( ? [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(replicate_int(Nb,Aa)))
          & aa_int_o(P,X) )
    <=> ( aa_int_o(P,Aa)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_5399_Bex__set__replicate,axiom,
    ! [Nb: nat,Aa: 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,Aa)))
          & aa_VEBT_VEBT_o(P,X) )
    <=> ( aa_VEBT_VEBT_o(P,Aa)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_5400_Ball__set__replicate,axiom,
    ! [Nb: nat,Aa: int,P: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(replicate_int(Nb,Aa)))
         => aa_int_o(P,X) )
    <=> ( aa_int_o(P,Aa)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_5401_Ball__set__replicate,axiom,
    ! [Nb: nat,Aa: 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,Aa)))
         => aa_VEBT_VEBT_o(P,X) )
    <=> ( aa_VEBT_VEBT_o(P,Aa)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_5402_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_5403_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_5404_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_5405_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_5406_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5407_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5408_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger((P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5409_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_5410_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_5411_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_5412_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_5413_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_5414_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_5415_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_5416_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gj),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5417_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gk),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5418_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_gl,unique3479559517661332726nteger(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5419_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5420_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(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,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5421_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5422_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5423_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5424_bits__1__div__exp,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5425_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_5426_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5427_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(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_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5428_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5429_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_5430_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_gm(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5431_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_gn(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5432_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_go(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5433_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_gp(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5434_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_gq(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5435_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_gr(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5436_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_gs(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5437_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_gt(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5438_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_gu(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5439_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_gv(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5440_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n3304061248610475627l_real(( (P)
        & (Q) )) = aa_real_real(times_times_real(zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q))) ).

% of_bool_conj
tff(fact_5441_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n2052037380579107095ol_rat(( (P)
        & (Q) )) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q))) ).

% of_bool_conj
tff(fact_5442_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_conj
tff(fact_5443_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        & (Q) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_conj
tff(fact_5444_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        & (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q))) ).

% of_bool_conj
tff(fact_5445_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_5446_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_5447_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_5448_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_5449_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_5450_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_5451_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_5452_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_5453_cond__case__prod__eta,axiom,
    ! [F: fun_na4138050610385705875at_nat,G: fun_Pr2118491707810114563at_nat] :
      ( ! [X2: nat,Y: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(F,X2),Y) = aa_Pro262675715343573244at_nat(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y))
     => ( produc2626176000494625587at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5454_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_o,G: fun_Pr4796943543331071514_nat_o] :
      ( ! [X2: nat,Y: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(F,X2),Y)
        <=> aa_Pro6150637604445964947_nat_o(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y)) )
     => ( produc6081775807080527818_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5455_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X2: int,Y: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X2),Y) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y))
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5456_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: int,Y: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X2),Y)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y)) )
     => ( produc4947309494688390418_int_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5457_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X2: int,Y: int] : aa_int_int(aa_int_fun_int_int(F,X2),Y) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y))
     => ( produc8211389475949308722nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5458_case__prod__eta,axiom,
    ! [F: fun_Pr2118491707810114563at_nat] : produc2626176000494625587at_nat(aTP_Lamm_gw(F)) = F ).

% case_prod_eta
tff(fact_5459_case__prod__eta,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o] : produc6081775807080527818_nat_o(aTP_Lamm_gx(F)) = F ).

% case_prod_eta
tff(fact_5460_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_gy(F)) = F ).

% case_prod_eta
tff(fact_5461_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_gz(F)) = F ).

% case_prod_eta
tff(fact_5462_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : produc8211389475949308722nt_int(aTP_Lamm_ha(F)) = F ).

% case_prod_eta
tff(fact_5463_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,Y: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y) )
           => ~ aa_Pro6150637604445964947_nat_o(Q,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(P,X2),Y)) ) ) ).

% case_prodE2
tff(fact_5464_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,Y: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y) )
           => ~ aa_o_o(Q,aa_nat_o(aa_nat_fun_nat_o(P,X2),Y)) ) ) ).

% case_prodE2
tff(fact_5465_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,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X2),Y)) ) ) ).

% case_prodE2
tff(fact_5466_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,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X2),Y)) ) ) ).

% case_prodE2
tff(fact_5467_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,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y) )
           => ~ aa_int_o(Q,aa_int_int(aa_int_fun_int_int(P,X2),Y)) ) ) ).

% case_prodE2
tff(fact_5468_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real((P))) ).

% zero_less_eq_of_bool
tff(fact_5469_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P))) ).

% zero_less_eq_of_bool
tff(fact_5470_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P))) ).

% zero_less_eq_of_bool
tff(fact_5471_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% zero_less_eq_of_bool
tff(fact_5472_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P))) ).

% zero_less_eq_of_bool
tff(fact_5473_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_5474_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_5475_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_5476_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_5477_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_5478_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P2)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_5479_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P2)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_5480_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P2)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_5481_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P2)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_5482_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P2)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_5483_split__of__bool__asm,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_Code_integer_o(P,one_one_Code_integer) )
          | ( ~ (P2)
            & ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_5484_split__of__bool,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ( ( (P2)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P2)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_5485_split__of__bool,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ( ( (P2)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P2)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_5486_split__of__bool,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ( ( (P2)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P2)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_5487_split__of__bool,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ( ( (P2)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P2)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_5488_split__of__bool,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ( ( (P2)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P2)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_5489_split__of__bool,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ( ( (P2)
         => aa_Code_integer_o(P,one_one_Code_integer) )
        & ( ~ (P2)
         => aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_5490_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n1201886186963655149omplex((P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_5491_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n3304061248610475627l_real((P2)) = $ite((P2),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_5492_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2052037380579107095ol_rat((P2)) = $ite((P2),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_5493_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,(P2)) = $ite((P2),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_5494_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P2)) = $ite((P2),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_5495_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n356916108424825756nteger((P2)) = $ite((P2),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_5496_replicate__eqI,axiom,
    ! [Xs: list_complex,Nb: nat,Xb: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Nb )
     => ( ! [Y: complex] :
            ( aa_set_complex_o(member_complex(Y),set_complex2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_complex(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5497_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,Xb: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y: real] :
            ( aa_set_real_o(member_real(Y),set_real2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_real(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5498_replicate__eqI,axiom,
    ! [Xs: list_set_nat,Nb: nat,Xb: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = Nb )
     => ( ! [Y: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(Y),set_set_nat2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_set_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5499_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,Xb: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y: nat] :
            ( aa_set_nat_o(member_nat(Y),set_nat2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5500_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y),set_VEBT_VEBT2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5501_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,Xb: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y: $o] :
            ( member_o((Y),set_o2(Xs))
           => ( (Y)
            <=> (Xb) ) )
       => ( Xs = replicate_o(Nb,(Xb)) ) ) ) ).

% replicate_eqI
tff(fact_5502_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,Xb: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y: int] :
            ( aa_set_int_o(member_int(Y),set_int2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_int(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5503_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_5504_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_5505_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_5506_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_5507_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_5508_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_5509_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_5510_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_5511_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_5512_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_5513_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_5514_of__bool__odd__eq__mod__2,axiom,
    ! [Aa: 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))),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_5515_of__bool__odd__eq__mod__2,axiom,
    ! [Aa: 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))),Aa)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_5516_of__bool__odd__eq__mod__2,axiom,
    ! [Aa: code_integer] : zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)) = modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_5517_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_5518_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Mb,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa) ) ).

% even_signed_take_bit_iff
tff(fact_5519_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,Aa: 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_ri631733984087533419it_int,Mb),Aa))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa) ) ).

% even_signed_take_bit_iff
tff(fact_5520_bits__induct,axiom,
    ! [P: fun_nat_o,Aa: nat] :
      ( ! [A2: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A2 )
         => aa_nat_o(P,A2) )
     => ( ! [A2: nat,B3: $o] :
            ( aa_nat_o(P,A2)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A2 )
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))) ) )
       => aa_nat_o(P,Aa) ) ) ).

% bits_induct
tff(fact_5521_bits__induct,axiom,
    ! [P: fun_int_o,Aa: int] :
      ( ! [A2: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A2 )
         => aa_int_o(P,A2) )
     => ( ! [A2: int,B3: $o] :
            ( aa_int_o(P,A2)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A2 )
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))) ) )
       => aa_int_o(P,Aa) ) ) ).

% bits_induct
tff(fact_5522_bits__induct,axiom,
    ! [P: fun_Code_integer_o,Aa: code_integer] :
      ( ! [A2: code_integer] :
          ( ( divide6298287555418463151nteger(A2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A2 )
         => aa_Code_integer_o(P,A2) )
     => ( ! [A2: code_integer,B3: $o] :
            ( aa_Code_integer_o(P,A2)
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A2)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A2 )
             => aa_Code_integer_o(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A2))) ) )
       => aa_Code_integer_o(P,Aa) ) ) ).

% bits_induct
tff(fact_5523_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_5524_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_5525_exp__mod__exp,axiom,
    ! [Mb: 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_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_5526_exp__mod__exp,axiom,
    ! [Mb: nat,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_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),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_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_5527_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_5528_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_5529_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(Xb,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5530_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_nat_nat] : size_option_nat(Xb,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5531_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_num_nat] : size_option_num(Xb,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5532_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_hb(L)),Qr) ).

% divmod_step_nat_def
tff(fact_5533_even__set__encode__iff,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( 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,A))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A) ) ) ).

% even_set_encode_iff
tff(fact_5534_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_5535_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),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_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),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,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_5536_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),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,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_5537_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_hc(L)),Qr) ).

% divmod_step_int_def
tff(fact_5538_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_5539_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_hb(L)),Qr) ).

% divmod_step_def
tff(fact_5540_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_hc(L)),Qr) ).

% divmod_step_def
tff(fact_5541_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_hd(L),Qr) ).

% divmod_step_def
tff(fact_5542_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Mb),Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ),
        aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),Mb),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_he),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ).

% divmod_nat_if
tff(fact_5543_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_5544_signed__take__bit__rec,axiom,
    ! [Nb: nat,Aa: code_integer] :
      bit_ri6519982836138164636nteger(Nb,Aa) = $ite(Nb = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(Aa,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))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5545_signed__take__bit__rec,axiom,
    ! [Nb: nat,Aa: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),Aa) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5546_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_5547_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y3: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y3 )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y3 = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y3 = 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)) )
                 => ( ( Y3 = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5548_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_5549_Compl__subset__Compl__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(A)),uminus1532241313380277803et_int(B2))
    <=> aa_set_int_o(ord_less_eq_set_int(B2),A) ) ).

% Compl_subset_Compl_iff
tff(fact_5550_Compl__anti__mono,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(B2)),uminus1532241313380277803et_int(A)) ) ).

% Compl_anti_mono
tff(fact_5551_case__prodI,axiom,
    ! [F: fun_Co480733329897759773un_o_o,Aa: code_integer,B: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(F,Aa),(B))
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(Aa,(B))) ) ).

% case_prodI
tff(fact_5552_case__prodI,axiom,
    ! [F: fun_num_fun_num_o,Aa: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(F,Aa),B)
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(Aa,B)) ) ).

% case_prodI
tff(fact_5553_case__prodI,axiom,
    ! [F: fun_nat_fun_num_o,Aa: nat,B: num] :
      ( aa_num_o(aa_nat_fun_num_o(F,Aa),B)
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(Aa,B)) ) ).

% case_prodI
tff(fact_5554_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_o,Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(F,Aa),B)
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Aa),B)) ) ).

% case_prodI
tff(fact_5555_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,Aa),B)
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Aa),B)) ) ).

% case_prodI
tff(fact_5556_case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,C: fun_Co480733329897759773un_o_o] :
      ( ! [A2: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A2,(B3)) )
         => aa_o_o(aa_Cod6379977994291289310un_o_o(C,A2),(B3)) )
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2) ) ).

% case_prodI2
tff(fact_5557_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: fun_num_fun_num_o] :
      ( ! [A2: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A2,B3) )
         => aa_num_o(aa_num_fun_num_o(C,A2),B3) )
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2) ) ).

% case_prodI2
tff(fact_5558_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: fun_nat_fun_num_o] :
      ( ! [A2: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A2,B3) )
         => aa_num_o(aa_nat_fun_num_o(C,A2),B3) )
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2) ) ).

% case_prodI2
tff(fact_5559_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A2: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A2),B3) )
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_5560_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A2: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B3) )
         => aa_int_o(aa_int_fun_int_o(C,A2),B3) )
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2) ) ).

% case_prodI2
tff(fact_5561_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_Co5015990272074005067omplex,Aa: code_integer,B: $o] :
      ( aa_set_complex_o(member_complex(Z),aa_o_set_complex(aa_Cod2779864889637189172omplex(C,Aa),(B)))
     => aa_set_complex_o(member_complex(Z),produc1043322548047392435omplex(C,produc6677183202524767010eger_o(Aa,(B)))) ) ).

% mem_case_prodI
tff(fact_5562_mem__case__prodI,axiom,
    ! [Z: real,C: fun_Co5368275265270938953t_real,Aa: code_integer,B: $o] :
      ( aa_set_real_o(member_real(Z),aa_o_set_real(aa_Cod7471715548787160626t_real(C,Aa),(B)))
     => aa_set_real_o(member_real(Z),produc242741666403216561t_real(C,produc6677183202524767010eger_o(Aa,(B)))) ) ).

% mem_case_prodI
tff(fact_5563_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_Co7401227239307276909et_nat,Aa: code_integer,B: $o] :
      ( aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,Aa),(B)))
     => aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,produc6677183202524767010eger_o(Aa,(B)))) ) ).

% mem_case_prodI
tff(fact_5564_mem__case__prodI,axiom,
    ! [Z: int,C: fun_Co2587678820519358153et_int,Aa: code_integer,B: $o] :
      ( aa_set_int_o(member_int(Z),aa_o_set_int(aa_Cod1361678317519647154et_int(C,Aa),(B)))
     => aa_set_int_o(member_int(Z),produc1253318751659547953et_int(C,produc6677183202524767010eger_o(Aa,(B)))) ) ).

% mem_case_prodI
tff(fact_5565_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,Aa: num,B: num] :
      ( aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,Aa),B))
     => aa_set_complex_o(member_complex(Z),produc2866383454006189126omplex(C,product_Pair_num_num(Aa,B))) ) ).

% mem_case_prodI
tff(fact_5566_mem__case__prodI,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,Aa: num,B: num] :
      ( aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,Aa),B))
     => aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,product_Pair_num_num(Aa,B))) ) ).

% mem_case_prodI
tff(fact_5567_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,Aa: num,B: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,Aa),B))
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,product_Pair_num_num(Aa,B))) ) ).

% mem_case_prodI
tff(fact_5568_mem__case__prodI,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,Aa: num,B: num] :
      ( aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,Aa),B))
     => aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,product_Pair_num_num(Aa,B))) ) ).

% mem_case_prodI
tff(fact_5569_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_na3925014923085203400omplex,Aa: nat,B: num] :
      ( aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,Aa),B))
     => aa_set_complex_o(member_complex(Z),produc6231982587499038204omplex(C,product_Pair_nat_num(Aa,B))) ) ).

% mem_case_prodI
tff(fact_5570_mem__case__prodI,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,Aa: nat,B: num] :
      ( aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,Aa),B))
     => aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,product_Pair_nat_num(Aa,B))) ) ).

% mem_case_prodI
tff(fact_5571_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: complex,C: fun_Co5015990272074005067omplex] :
      ( ! [A2: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A2,(B3)) )
         => aa_set_complex_o(member_complex(Z),aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A2),(B3))) )
     => aa_set_complex_o(member_complex(Z),produc1043322548047392435omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5572_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: real,C: fun_Co5368275265270938953t_real] :
      ( ! [A2: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A2,(B3)) )
         => aa_set_real_o(member_real(Z),aa_o_set_real(aa_Cod7471715548787160626t_real(C,A2),(B3))) )
     => aa_set_real_o(member_real(Z),produc242741666403216561t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5573_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: nat,C: fun_Co7401227239307276909et_nat] :
      ( ! [A2: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A2,(B3)) )
         => aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A2),(B3))) )
     => aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5574_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: int,C: fun_Co2587678820519358153et_int] :
      ( ! [A2: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A2,(B3)) )
         => aa_set_int_o(member_int(Z),aa_o_set_int(aa_Cod1361678317519647154et_int(C,A2),(B3))) )
     => aa_set_int_o(member_int(Z),produc1253318751659547953et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5575_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: complex,C: fun_nu9041852840018610706omplex] :
      ( ! [A2: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A2,B3) )
         => aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,A2),B3)) )
     => aa_set_complex_o(member_complex(Z),produc2866383454006189126omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5576_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: real,C: fun_nu4698968662094489488t_real] :
      ( ! [A2: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A2,B3) )
         => aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,A2),B3)) )
     => aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5577_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: nat,C: fun_nu530747996811530292et_nat] :
      ( ! [A2: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A2,B3) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A2),B3)) )
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5578_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: int,C: fun_nu4940571614878387344et_int] :
      ( ! [A2: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A2,B3) )
         => aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,A2),B3)) )
     => aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5579_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: complex,C: fun_na3925014923085203400omplex] :
      ( ! [A2: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A2,B3) )
         => aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,A2),B3)) )
     => aa_set_complex_o(member_complex(Z),produc6231982587499038204omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5580_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: real,C: fun_na3666575854997362246t_real] :
      ( ! [A2: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A2,B3) )
         => aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,A2),B3)) )
     => aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5581_neg__le__iff__le,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ).

% neg_le_iff_le
tff(fact_5582_neg__le__iff__le,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),B) ) ).

% neg_le_iff_le
tff(fact_5583_neg__le__iff__le,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ).

% neg_le_iff_le
tff(fact_5584_neg__le__iff__le,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B) ) ).

% neg_le_iff_le
tff(fact_5585_neg__less__iff__less,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B) ) ).

% neg_less_iff_less
tff(fact_5586_neg__less__iff__less,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ).

% neg_less_iff_less
tff(fact_5587_neg__less__iff__less,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B) ) ).

% neg_less_iff_less
tff(fact_5588_neg__less__iff__less,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),B) ) ).

% neg_less_iff_less
tff(fact_5589_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5590_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5591_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5592_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5593_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5594_mult__minus__left,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,Aa)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B)) ).

% mult_minus_left
tff(fact_5595_mult__minus__left,axiom,
    ! [Aa: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,Aa)),B) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(Aa),B)) ).

% mult_minus_left
tff(fact_5596_mult__minus__left,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B)) ).

% mult_minus_left
tff(fact_5597_mult__minus__left,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,Aa)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(Aa),B)) ).

% mult_minus_left
tff(fact_5598_mult__minus__left,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B)) ).

% mult_minus_left
tff(fact_5599_minus__mult__minus,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B) ).

% minus_mult_minus
tff(fact_5600_minus__mult__minus,axiom,
    ! [Aa: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,Aa)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(times_times_real(Aa),B) ).

% minus_mult_minus
tff(fact_5601_minus__mult__minus,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B) ).

% minus_mult_minus
tff(fact_5602_minus__mult__minus,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(times_times_rat(Aa),B) ).

% minus_mult_minus
tff(fact_5603_minus__mult__minus,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B) ).

% minus_mult_minus
tff(fact_5604_mult__minus__right,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B)) ).

% mult_minus_right
tff(fact_5605_mult__minus__right,axiom,
    ! [Aa: real,B: real] : aa_real_real(times_times_real(Aa),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(Aa),B)) ).

% mult_minus_right
tff(fact_5606_mult__minus__right,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),B)) ).

% mult_minus_right
tff(fact_5607_mult__minus__right,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(Aa),B)) ).

% mult_minus_right
tff(fact_5608_mult__minus__right,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B)) ).

% mult_minus_right
tff(fact_5609_minus__add__distrib,axiom,
    ! [Aa: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_5610_minus__add__distrib,axiom,
    ! [Aa: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(Aa),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Aa)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_5611_minus__add__distrib,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_5612_minus__add__distrib,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(Aa),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_5613_minus__add__distrib,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(Aa,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_5614_minus__add__cancel,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B)) = B ).

% minus_add_cancel
tff(fact_5615_minus__add__cancel,axiom,
    ! [Aa: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Aa)),aa_real_real(plus_plus_real(Aa),B)) = B ).

% minus_add_cancel
tff(fact_5616_minus__add__cancel,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B)) = B ).

% minus_add_cancel
tff(fact_5617_minus__add__cancel,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(plus_plus_rat(Aa),B)) = B ).

% minus_add_cancel
tff(fact_5618_minus__add__cancel,axiom,
    ! [Aa: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),plus_p5714425477246183910nteger(Aa,B)) = B ).

% minus_add_cancel
tff(fact_5619_add__minus__cancel,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),B)) = B ).

% add_minus_cancel
tff(fact_5620_add__minus__cancel,axiom,
    ! [Aa: real,B: real] : aa_real_real(plus_plus_real(Aa),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Aa)),B)) = B ).

% add_minus_cancel
tff(fact_5621_add__minus__cancel,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),B)) = B ).

% add_minus_cancel
tff(fact_5622_add__minus__cancel,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),B)) = B ).

% add_minus_cancel
tff(fact_5623_add__minus__cancel,axiom,
    ! [Aa: code_integer,B: code_integer] : plus_p5714425477246183910nteger(Aa,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),B)) = B ).

% add_minus_cancel
tff(fact_5624_div__minus__minus,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B) ).

% div_minus_minus
tff(fact_5625_div__minus__minus,axiom,
    ! [Aa: code_integer,B: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(Aa,B) ).

% div_minus_minus
tff(fact_5626_mod__minus__minus,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B)) ).

% mod_minus_minus
tff(fact_5627_mod__minus__minus,axiom,
    ! [Aa: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(Aa,B)) ).

% mod_minus_minus
tff(fact_5628_real__add__minus__iff,axiom,
    ! [Xb: real,Aa: real] :
      ( ( aa_real_real(plus_plus_real(Xb),aa_real_real(uminus_uminus_real,Aa)) = zero_zero_real )
    <=> ( Xb = Aa ) ) ).

% real_add_minus_iff
tff(fact_5629_neg__0__le__iff__le,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_5630_neg__0__le__iff__le,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_5631_neg__0__le__iff__le,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_5632_neg__0__le__iff__le,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_5633_neg__le__0__iff__le,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% neg_le_0_iff_le
tff(fact_5634_neg__le__0__iff__le,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa) ) ).

% neg_le_0_iff_le
tff(fact_5635_neg__le__0__iff__le,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% neg_le_0_iff_le
tff(fact_5636_neg__le__0__iff__le,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ).

% neg_le_0_iff_le
tff(fact_5637_less__eq__neg__nonpos,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_5638_less__eq__neg__nonpos,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_5639_less__eq__neg__nonpos,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_5640_less__eq__neg__nonpos,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_5641_neg__less__eq__nonneg,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% neg_less_eq_nonneg
tff(fact_5642_neg__less__eq__nonneg,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Aa)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa) ) ).

% neg_less_eq_nonneg
tff(fact_5643_neg__less__eq__nonneg,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% neg_less_eq_nonneg
tff(fact_5644_neg__less__eq__nonneg,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),Aa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ).

% neg_less_eq_nonneg
tff(fact_5645_neg__less__0__iff__less,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Aa)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ).

% neg_less_0_iff_less
tff(fact_5646_neg__less__0__iff__less,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ).

% neg_less_0_iff_less
tff(fact_5647_neg__less__0__iff__less,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ).

% neg_less_0_iff_less
tff(fact_5648_neg__less__0__iff__less,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Aa) ) ).

% neg_less_0_iff_less
tff(fact_5649_neg__0__less__iff__less,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_5650_neg__0__less__iff__less,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_5651_neg__0__less__iff__less,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_5652_neg__0__less__iff__less,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_5653_neg__less__pos,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Aa)),Aa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ).

% neg_less_pos
tff(fact_5654_neg__less__pos,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Aa)),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ).

% neg_less_pos
tff(fact_5655_neg__less__pos,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ).

% neg_less_pos
tff(fact_5656_neg__less__pos,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Aa)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Aa) ) ).

% neg_less_pos
tff(fact_5657_less__neg__neg,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% less_neg_neg
tff(fact_5658_less__neg__neg,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% less_neg_neg
tff(fact_5659_less__neg__neg,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_5660_less__neg__neg,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_5661_ab__left__minus,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),Aa) = zero_zero_int ).

% ab_left_minus
tff(fact_5662_ab__left__minus,axiom,
    ! [Aa: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Aa)),Aa) = zero_zero_real ).

% ab_left_minus
tff(fact_5663_ab__left__minus,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),Aa) = zero_zero_complex ).

% ab_left_minus
tff(fact_5664_ab__left__minus,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Aa) = zero_zero_rat ).

% ab_left_minus
tff(fact_5665_ab__left__minus,axiom,
    ! [Aa: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),Aa) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_5666_add_Oright__inverse,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(uminus_uminus_int,Aa)) = zero_zero_int ).

% add.right_inverse
tff(fact_5667_add_Oright__inverse,axiom,
    ! [Aa: real] : aa_real_real(plus_plus_real(Aa),aa_real_real(uminus_uminus_real,Aa)) = zero_zero_real ).

% add.right_inverse
tff(fact_5668_add_Oright__inverse,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),aa_complex_complex(uminus1482373934393186551omplex,Aa)) = zero_zero_complex ).

% add.right_inverse
tff(fact_5669_add_Oright__inverse,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(uminus_uminus_rat,Aa)) = zero_zero_rat ).

% add.right_inverse
tff(fact_5670_add_Oright__inverse,axiom,
    ! [Aa: code_integer] : plus_p5714425477246183910nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_5671_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5672_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5673_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5674_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5675_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5676_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_5677_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(times_times_real(Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_5678_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_5679_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_5680_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_5681_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_5682_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_5683_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_5684_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_5685_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_5686_divide__minus1,axiom,
    ! [Xb: real] : aa_real_real(divide_divide_real(Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xb) ).

% divide_minus1
tff(fact_5687_divide__minus1,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% divide_minus1
tff(fact_5688_divide__minus1,axiom,
    ! [Xb: rat] : aa_rat_rat(divide_divide_rat(Xb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Xb) ).

% divide_minus1
tff(fact_5689_div__minus1__right,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Aa) ).

% div_minus1_right
tff(fact_5690_div__minus1__right,axiom,
    ! [Aa: code_integer] : divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ).

% div_minus1_right
tff(fact_5691_diff__minus__eq__add,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) ).

% diff_minus_eq_add
tff(fact_5692_diff__minus__eq__add,axiom,
    ! [Aa: real,B: real] : aa_real_real(minus_minus_real(Aa),aa_real_real(uminus_uminus_real,B)) = aa_real_real(plus_plus_real(Aa),B) ).

% diff_minus_eq_add
tff(fact_5693_diff__minus__eq__add,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B) ).

% diff_minus_eq_add
tff(fact_5694_diff__minus__eq__add,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(Aa),B) ).

% diff_minus_eq_add
tff(fact_5695_diff__minus__eq__add,axiom,
    ! [Aa: code_integer,B: code_integer] : minus_8373710615458151222nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(Aa,B) ).

% diff_minus_eq_add
tff(fact_5696_uminus__add__conv__diff,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Aa) ).

% uminus_add_conv_diff
tff(fact_5697_uminus__add__conv__diff,axiom,
    ! [Aa: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Aa)),B) = aa_real_real(minus_minus_real(B),Aa) ).

% uminus_add_conv_diff
tff(fact_5698_uminus__add__conv__diff,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),B) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),Aa) ).

% uminus_add_conv_diff
tff(fact_5699_uminus__add__conv__diff,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),B) = aa_rat_rat(minus_minus_rat(B),Aa) ).

% uminus_add_conv_diff
tff(fact_5700_uminus__add__conv__diff,axiom,
    ! [Aa: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),B) = minus_8373710615458151222nteger(B,Aa) ).

% uminus_add_conv_diff
tff(fact_5701_minus__mod__self1,axiom,
    ! [B: int,Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Aa)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),B) ).

% minus_mod_self1
tff(fact_5702_minus__mod__self1,axiom,
    ! [B: code_integer,Aa: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,Aa),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),B) ).

% minus_mod_self1
tff(fact_5703_subset__Compl__singleton,axiom,
    ! [A: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),uminus8041839845116263051T_VEBT(insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT)))
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),A) ) ).

% subset_Compl_singleton
tff(fact_5704_subset__Compl__singleton,axiom,
    ! [A: set_complex,B: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),uminus8566677241136511917omplex(insert_complex(B,bot_bot_set_complex)))
    <=> ~ aa_set_complex_o(member_complex(B),A) ) ).

% subset_Compl_singleton
tff(fact_5705_subset__Compl__singleton,axiom,
    ! [A: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat)))
    <=> ~ aa_set_set_nat_o(member_set_nat(B),A) ) ).

% subset_Compl_singleton
tff(fact_5706_subset__Compl__singleton,axiom,
    ! [A: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B),A) ) ).

% subset_Compl_singleton
tff(fact_5707_subset__Compl__singleton,axiom,
    ! [A: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),uminus612125837232591019t_real(insert_real(B,bot_bot_set_real)))
    <=> ~ aa_set_real_o(member_real(B),A) ) ).

% subset_Compl_singleton
tff(fact_5708_subset__Compl__singleton,axiom,
    ! [A: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B),A) ) ).

% subset_Compl_singleton
tff(fact_5709_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : bit_ri6519982836138164636nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_5710_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_5711_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_5712_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_5713_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_5714_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_5715_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_simps(1)
tff(fact_5716_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_5717_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_simps(1)
tff(fact_5718_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_simps(1)
tff(fact_5719_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_5720_dbl__inc__simps_I4_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_inc_simps(4)
tff(fact_5721_dbl__inc__simps_I4_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_inc_simps(4)
tff(fact_5722_dbl__inc__simps_I4_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_5723_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_5724_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_5725_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_5726_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_5727_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_5728_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_5729_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_5730_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_5731_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_5732_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_5733_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_5734_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5735_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5736_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5737_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5738_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5739_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5740_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5741_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5742_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5743_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5744_diff__numeral__special_I12_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_5745_diff__numeral__special_I12_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_5746_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_5747_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_5748_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_5749_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_5750_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)) = one_one_real ).

% minus_one_mult_self
tff(fact_5751_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)) = one_one_complex ).

% minus_one_mult_self
tff(fact_5752_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)) = one_one_rat ).

% minus_one_mult_self
tff(fact_5753_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_5754_left__minus__one__mult__self,axiom,
    ! [Nb: nat,Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_5755_left__minus__one__mult__self,axiom,
    ! [Nb: nat,Aa: real] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_5756_left__minus__one__mult__self,axiom,
    ! [Nb: nat,Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_5757_left__minus__one__mult__self,axiom,
    ! [Nb: nat,Aa: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_5758_left__minus__one__mult__self,axiom,
    ! [Nb: nat,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_5759_mod__minus1__right,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_5760_mod__minus1__right,axiom,
    ! [Aa: code_integer] : modulo364778990260209775nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_5761_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(2)
tff(fact_5762_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,U)) ).

% max_number_of(2)
tff(fact_5763_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(2)
tff(fact_5764_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_5765_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(3)
tff(fact_5766_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(3)
tff(fact_5767_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(3)
tff(fact_5768_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_5769_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(4)
tff(fact_5770_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(4)
tff(fact_5771_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(4)
tff(fact_5772_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_5773_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y3) ).

% semiring_norm(168)
tff(fact_5774_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y3)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y3) ).

% semiring_norm(168)
tff(fact_5775_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y3) ).

% semiring_norm(168)
tff(fact_5776_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y3)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y3) ).

% semiring_norm(168)
tff(fact_5777_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y3)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y3) ).

% semiring_norm(168)
tff(fact_5778_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5779_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5780_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5781_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5782_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5783_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5784_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5785_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5786_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5787_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5788_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5789_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5790_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5791_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5792_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5793_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5794_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5795_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5796_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5797_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5798_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5799_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5800_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5801_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5802_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5803_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y3)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(170)
tff(fact_5804_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),Y3)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(170)
tff(fact_5805_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(170)
tff(fact_5806_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Y3)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(170)
tff(fact_5807_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(170)
tff(fact_5808_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y3)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(171)
tff(fact_5809_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y3)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(171)
tff(fact_5810_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(171)
tff(fact_5811_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y3)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(171)
tff(fact_5812_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y3) ).

% semiring_norm(171)
tff(fact_5813_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y3)) = 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))),Y3) ).

% semiring_norm(172)
tff(fact_5814_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y3)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y3) ).

% semiring_norm(172)
tff(fact_5815_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y3) ).

% semiring_norm(172)
tff(fact_5816_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y3)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y3) ).

% semiring_norm(172)
tff(fact_5817_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y3) ).

% semiring_norm(172)
tff(fact_5818_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_5819_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_5820_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_5821_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_5822_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_5823_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_5824_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_5825_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5826_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5827_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5828_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5829_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5830_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5831_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5832_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5833_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_5834_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_5835_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_5836_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5837_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5838_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5839_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5840_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5841_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5842_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5843_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5844_le__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5845_le__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5846_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5847_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5848_eq__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: real,B: real,W: num] :
      ( ( Aa = aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,aa_real_real(times_times_real(Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B,Aa = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5849_eq__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: complex,B: complex,W: num] :
      ( ( Aa = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B,Aa = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5850_eq__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: rat,B: rat,W: num] :
      ( ( Aa = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B,Aa = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5851_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,Aa: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = Aa )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,B = aa_real_real(times_times_real(Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Aa = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5852_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,Aa: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = Aa )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Aa = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5853_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = Aa )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Aa = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5854_less__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5855_less__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5856_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5857_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5858_power2__minus,axiom,
    ! [Aa: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5859_power2__minus,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5860_power2__minus,axiom,
    ! [Aa: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5861_power2__minus,axiom,
    ! [Aa: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5862_power2__minus,axiom,
    ! [Aa: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5863_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5864_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5865_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5866_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5867_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5868_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5869_diff__numeral__special_I10_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5870_diff__numeral__special_I10_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5871_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5872_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5873_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5874_diff__numeral__special_I11_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5875_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5876_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5877_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5878_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_5879_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_5880_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_5881_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_5882_minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_5883_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_5884_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [Aa: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Aa)),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),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5885_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [Aa: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Aa)),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),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5886_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [Aa: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5887_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [Aa: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),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(power_power_rat(Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5888_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [Aa: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),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_Code_integer(power_8256067586552552935nteger(Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5889_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,Aa: 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_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Aa)),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5890_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,Aa: 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(uminus_uminus_real,Aa)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5891_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,Aa: complex] :
      ( 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_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),Nb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5892_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,Aa: 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_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Nb) = aa_nat_rat(power_power_rat(Aa),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5893_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,Aa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5894_power__minus__odd,axiom,
    ! [Nb: nat,Aa: 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_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Aa)),Nb) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ) ) ).

% power_minus_odd
tff(fact_5895_power__minus__odd,axiom,
    ! [Nb: nat,Aa: 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(uminus_uminus_real,Aa)),Nb) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ) ) ).

% power_minus_odd
tff(fact_5896_power__minus__odd,axiom,
    ! [Nb: nat,Aa: complex] :
      ( ~ 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_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)) ) ) ).

% power_minus_odd
tff(fact_5897_power__minus__odd,axiom,
    ! [Nb: nat,Aa: 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_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Nb) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Aa),Nb)) ) ) ).

% power_minus_odd
tff(fact_5898_power__minus__odd,axiom,
    ! [Nb: nat,Aa: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb)) ) ) ).

% power_minus_odd
tff(fact_5899_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5900_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5901_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5902_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5903_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5904_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5905_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5906_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5907_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5908_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5909_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_5910_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5911_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5912_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5913_dbl__simps_I4_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5914_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5915_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_int ).

% power_minus1_even
tff(fact_5916_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_real ).

% power_minus1_even
tff(fact_5917_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),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_complex ).

% power_minus1_even
tff(fact_5918_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_rat ).

% power_minus1_even
tff(fact_5919_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_5920_neg__one__even__power,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_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_5921_neg__one__even__power,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_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_5922_neg__one__even__power,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_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_5923_neg__one__even__power,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_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_5924_neg__one__even__power,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_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_5925_neg__one__odd__power,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_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_5926_neg__one__odd__power,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_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_5927_neg__one__odd__power,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_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_5928_neg__one__odd__power,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_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_5929_neg__one__odd__power,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_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_5930_signed__take__bit__0,axiom,
    ! [Aa: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,Aa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5931_signed__take__bit__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,zero_zero_nat),Aa) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5932_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_5933_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_5934_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_5935_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_5936_signed__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(uminus_uminus_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),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_5937_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_Co5015990272074005067omplex,P2: produc6271795597528267376eger_o] :
      ( aa_set_complex_o(member_complex(Z),produc1043322548047392435omplex(C,P2))
     => ~ ! [X2: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y)) )
           => ~ aa_set_complex_o(member_complex(Z),aa_o_set_complex(aa_Cod2779864889637189172omplex(C,X2),(Y))) ) ) ).

% mem_case_prodE
tff(fact_5938_mem__case__prodE,axiom,
    ! [Z: real,C: fun_Co5368275265270938953t_real,P2: produc6271795597528267376eger_o] :
      ( aa_set_real_o(member_real(Z),produc242741666403216561t_real(C,P2))
     => ~ ! [X2: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y)) )
           => ~ aa_set_real_o(member_real(Z),aa_o_set_real(aa_Cod7471715548787160626t_real(C,X2),(Y))) ) ) ).

% mem_case_prodE
tff(fact_5939_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_Co7401227239307276909et_nat,P2: produc6271795597528267376eger_o] :
      ( aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2))
     => ~ ! [X2: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y)) )
           => ~ aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,X2),(Y))) ) ) ).

% mem_case_prodE
tff(fact_5940_mem__case__prodE,axiom,
    ! [Z: int,C: fun_Co2587678820519358153et_int,P2: produc6271795597528267376eger_o] :
      ( aa_set_int_o(member_int(Z),produc1253318751659547953et_int(C,P2))
     => ~ ! [X2: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y)) )
           => ~ aa_set_int_o(member_int(Z),aa_o_set_int(aa_Cod1361678317519647154et_int(C,X2),(Y))) ) ) ).

% mem_case_prodE
tff(fact_5941_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,P2: product_prod_num_num] :
      ( aa_set_complex_o(member_complex(Z),produc2866383454006189126omplex(C,P2))
     => ~ ! [X2: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X2,Y) )
           => ~ aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,X2),Y)) ) ) ).

% mem_case_prodE
tff(fact_5942_mem__case__prodE,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,P2: product_prod_num_num] :
      ( aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,P2))
     => ~ ! [X2: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X2,Y) )
           => ~ aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,X2),Y)) ) ) ).

% mem_case_prodE
tff(fact_5943_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,P2: product_prod_num_num] :
      ( aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2))
     => ~ ! [X2: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X2,Y) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,X2),Y)) ) ) ).

% mem_case_prodE
tff(fact_5944_mem__case__prodE,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,P2: product_prod_num_num] :
      ( aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,P2))
     => ~ ! [X2: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X2,Y) )
           => ~ aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,X2),Y)) ) ) ).

% mem_case_prodE
tff(fact_5945_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_na3925014923085203400omplex,P2: product_prod_nat_num] :
      ( aa_set_complex_o(member_complex(Z),produc6231982587499038204omplex(C,P2))
     => ~ ! [X2: nat,Y: num] :
            ( ( P2 = product_Pair_nat_num(X2,Y) )
           => ~ aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,X2),Y)) ) ) ).

% mem_case_prodE
tff(fact_5946_mem__case__prodE,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,P2: product_prod_nat_num] :
      ( aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,P2))
     => ~ ! [X2: nat,Y: num] :
            ( ( P2 = product_Pair_nat_num(X2,Y) )
           => ~ aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,X2),Y)) ) ) ).

% mem_case_prodE
tff(fact_5947_le__imp__neg__le,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,Aa)) ) ).

% le_imp_neg_le
tff(fact_5948_le__imp__neg__le,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ).

% le_imp_neg_le
tff(fact_5949_le__imp__neg__le,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,Aa)) ) ).

% le_imp_neg_le
tff(fact_5950_le__imp__neg__le,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,Aa)) ) ).

% le_imp_neg_le
tff(fact_5951_minus__le__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),Aa) ) ).

% minus_le_iff
tff(fact_5952_minus__le__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),Aa) ) ).

% minus_le_iff
tff(fact_5953_minus__le__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),Aa) ) ).

% minus_le_iff
tff(fact_5954_minus__le__iff,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),Aa) ) ).

% minus_le_iff
tff(fact_5955_le__minus__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,Aa)) ) ).

% le_minus_iff
tff(fact_5956_le__minus__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ).

% le_minus_iff
tff(fact_5957_le__minus__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,Aa)) ) ).

% le_minus_iff
tff(fact_5958_le__minus__iff,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,Aa)) ) ).

% le_minus_iff
tff(fact_5959_less__minus__iff,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,Aa)) ) ).

% less_minus_iff
tff(fact_5960_less__minus__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,Aa)) ) ).

% less_minus_iff
tff(fact_5961_less__minus__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,Aa)) ) ).

% less_minus_iff
tff(fact_5962_less__minus__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ).

% less_minus_iff
tff(fact_5963_minus__less__iff,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Aa)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),Aa) ) ).

% minus_less_iff
tff(fact_5964_minus__less__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Aa)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),Aa) ) ).

% minus_less_iff
tff(fact_5965_minus__less__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Aa)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),Aa) ) ).

% minus_less_iff
tff(fact_5966_minus__less__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),Aa) ) ).

% minus_less_iff
tff(fact_5967_verit__negate__coefficient_I2_J,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,Aa)) ) ).

% verit_negate_coefficient(2)
tff(fact_5968_verit__negate__coefficient_I2_J,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,Aa)) ) ).

% verit_negate_coefficient(2)
tff(fact_5969_verit__negate__coefficient_I2_J,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,Aa)) ) ).

% verit_negate_coefficient(2)
tff(fact_5970_verit__negate__coefficient_I2_J,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ).

% verit_negate_coefficient(2)
tff(fact_5971_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,Mb) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5972_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_real(numeral_numeral_real,Mb) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5973_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numera6690914467698888265omplex(Mb) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5974_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_rat(numeral_numeral_rat,Mb) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5975_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_Code_integer(numera6620942414471956472nteger,Mb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5976_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5977_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) != aa_num_real(numeral_numeral_real,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5978_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5979_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) != aa_num_rat(numeral_numeral_rat,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5980_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)) != aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5981_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_5982_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_5983_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_5984_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_5985_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_5986_square__eq__iff,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Aa) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( Aa = B )
        | ( Aa = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_5987_square__eq__iff,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(times_times_real(Aa),Aa) = aa_real_real(times_times_real(B),B) )
    <=> ( ( Aa = B )
        | ( Aa = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_5988_square__eq__iff,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),Aa) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),B) )
    <=> ( ( Aa = B )
        | ( Aa = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_5989_square__eq__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),Aa) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( Aa = B )
        | ( Aa = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_5990_square__eq__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( Aa = B )
        | ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_5991_minus__mult__commute,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,Aa)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_5992_minus__mult__commute,axiom,
    ! [Aa: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,Aa)),B) = aa_real_real(times_times_real(Aa),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_5993_minus__mult__commute,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),B) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_5994_minus__mult__commute,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,Aa)),B) = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_5995_minus__mult__commute,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_5996_is__num__normalize_I8_J,axiom,
    ! [Aa: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,Aa)) ).

% is_num_normalize(8)
tff(fact_5997_is__num__normalize_I8_J,axiom,
    ! [Aa: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(Aa),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,Aa)) ).

% is_num_normalize(8)
tff(fact_5998_is__num__normalize_I8_J,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,Aa)) ).

% is_num_normalize(8)
tff(fact_5999_is__num__normalize_I8_J,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(Aa),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,Aa)) ).

% is_num_normalize(8)
tff(fact_6000_is__num__normalize_I8_J,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(Aa,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ).

% is_num_normalize(8)
tff(fact_6001_add_Oinverse__distrib__swap,axiom,
    ! [Aa: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,Aa)) ).

% add.inverse_distrib_swap
tff(fact_6002_add_Oinverse__distrib__swap,axiom,
    ! [Aa: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(Aa),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,Aa)) ).

% add.inverse_distrib_swap
tff(fact_6003_add_Oinverse__distrib__swap,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,Aa)) ).

% add.inverse_distrib_swap
tff(fact_6004_add_Oinverse__distrib__swap,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(Aa),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,Aa)) ).

% add.inverse_distrib_swap
tff(fact_6005_add_Oinverse__distrib__swap,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(Aa,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ).

% add.inverse_distrib_swap
tff(fact_6006_group__cancel_Oneg1,axiom,
    ! [A: int,K: int,Aa: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),Aa) )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,Aa)) ) ) ).

% group_cancel.neg1
tff(fact_6007_group__cancel_Oneg1,axiom,
    ! [A: real,K: real,Aa: real] :
      ( ( A = aa_real_real(plus_plus_real(K),Aa) )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,Aa)) ) ) ).

% group_cancel.neg1
tff(fact_6008_group__cancel_Oneg1,axiom,
    ! [A: complex,K: complex,Aa: complex] :
      ( ( A = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,K),Aa) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,Aa)) ) ) ).

% group_cancel.neg1
tff(fact_6009_group__cancel_Oneg1,axiom,
    ! [A: rat,K: rat,Aa: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(K),Aa) )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,Aa)) ) ) ).

% group_cancel.neg1
tff(fact_6010_group__cancel_Oneg1,axiom,
    ! [A: code_integer,K: code_integer,Aa: code_integer] :
      ( ( A = plus_p5714425477246183910nteger(K,Aa) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ) ).

% group_cancel.neg1
tff(fact_6011_minus__diff__minus,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B)) ).

% minus_diff_minus
tff(fact_6012_minus__diff__minus,axiom,
    ! [Aa: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Aa)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(Aa),B)) ).

% minus_diff_minus
tff(fact_6013_minus__diff__minus,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),B)) ).

% minus_diff_minus
tff(fact_6014_minus__diff__minus,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(Aa),B)) ).

% minus_diff_minus
tff(fact_6015_minus__diff__minus,axiom,
    ! [Aa: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(Aa,B)) ).

% minus_diff_minus
tff(fact_6016_minus__divide__left,axiom,
    ! [Aa: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Aa),B)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,Aa)),B) ).

% minus_divide_left
tff(fact_6017_minus__divide__left,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),B) ).

% minus_divide_left
tff(fact_6018_minus__divide__left,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,Aa)),B) ).

% minus_divide_left
tff(fact_6019_minus__divide__divide,axiom,
    ! [Aa: real,B: real] : aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,Aa)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(Aa),B) ).

% minus_divide_divide
tff(fact_6020_minus__divide__divide,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) ).

% minus_divide_divide
tff(fact_6021_minus__divide__divide,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(Aa),B) ).

% minus_divide_divide
tff(fact_6022_minus__divide__right,axiom,
    ! [Aa: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Aa),B)) = aa_real_real(divide_divide_real(Aa),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_6023_minus__divide__right,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_6024_minus__divide__right,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),B)) = aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_divide_right
tff(fact_6025_div__minus__right,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,Aa)),B) ).

% div_minus_right
tff(fact_6026_div__minus__right,axiom,
    ! [Aa: code_integer,B: code_integer] : divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),B) ).

% div_minus_right
tff(fact_6027_mod__minus__right,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),B)) ).

% mod_minus_right
tff(fact_6028_mod__minus__right,axiom,
    ! [Aa: code_integer,B: code_integer] : modulo364778990260209775nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),B)) ).

% mod_minus_right
tff(fact_6029_mod__minus__cong,axiom,
    ! [Aa: int,B: int,A4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),B) )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A4)),B) ) ) ).

% mod_minus_cong
tff(fact_6030_mod__minus__cong,axiom,
    ! [Aa: code_integer,B: code_integer,A4: code_integer] :
      ( ( modulo364778990260209775nteger(Aa,B) = modulo364778990260209775nteger(A4,B) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A4),B) ) ) ).

% mod_minus_cong
tff(fact_6031_mod__minus__eq,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),B) ).

% mod_minus_eq
tff(fact_6032_mod__minus__eq,axiom,
    ! [Aa: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(Aa,B)),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),B) ).

% mod_minus_eq
tff(fact_6033_case__prodD,axiom,
    ! [F: fun_Co480733329897759773un_o_o,Aa: code_integer,B: $o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(Aa,(B)))
     => aa_o_o(aa_Cod6379977994291289310un_o_o(F,Aa),(B)) ) ).

% case_prodD
tff(fact_6034_case__prodD,axiom,
    ! [F: fun_num_fun_num_o,Aa: num,B: num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(Aa,B))
     => aa_num_o(aa_num_fun_num_o(F,Aa),B) ) ).

% case_prodD
tff(fact_6035_case__prodD,axiom,
    ! [F: fun_nat_fun_num_o,Aa: nat,B: num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(Aa,B))
     => aa_num_o(aa_nat_fun_num_o(F,Aa),B) ) ).

% case_prodD
tff(fact_6036_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_o,Aa: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Aa),B))
     => aa_nat_o(aa_nat_fun_nat_o(F,Aa),B) ) ).

% case_prodD
tff(fact_6037_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,Aa: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Aa),B))
     => aa_int_o(aa_int_fun_int_o(F,Aa),B) ) ).

% case_prodD
tff(fact_6038_case__prodE,axiom,
    ! [C: fun_Co480733329897759773un_o_o,P2: produc6271795597528267376eger_o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2)
     => ~ ! [X2: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y)) )
           => ~ aa_o_o(aa_Cod6379977994291289310un_o_o(C,X2),(Y)) ) ) ).

% case_prodE
tff(fact_6039_case__prodE,axiom,
    ! [C: fun_num_fun_num_o,P2: product_prod_num_num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2)
     => ~ ! [X2: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X2,Y) )
           => ~ aa_num_o(aa_num_fun_num_o(C,X2),Y) ) ) ).

% case_prodE
tff(fact_6040_case__prodE,axiom,
    ! [C: fun_nat_fun_num_o,P2: product_prod_nat_num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2)
     => ~ ! [X2: nat,Y: num] :
            ( ( P2 = product_Pair_nat_num(X2,Y) )
           => ~ aa_num_o(aa_nat_fun_num_o(C,X2),Y) ) ) ).

% case_prodE
tff(fact_6041_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,Y: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X2),Y) ) ) ).

% case_prodE
tff(fact_6042_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,Y: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X2),Y) ) ) ).

% case_prodE
tff(fact_6043_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_6044_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_6045_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_6046_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_6047_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_le_numeral
tff(fact_6048_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% neg_numeral_le_numeral
tff(fact_6049_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_le_numeral
tff(fact_6050_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_6051_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% zero_neq_neg_numeral
tff(fact_6052_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% zero_neq_neg_numeral
tff(fact_6053_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_6054_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_neq_neg_numeral
tff(fact_6055_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% zero_neq_neg_numeral
tff(fact_6056_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_6057_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_6058_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_6059_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_6060_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_6061_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_less_numeral
tff(fact_6062_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_less_numeral
tff(fact_6063_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% neg_numeral_less_numeral
tff(fact_6064_le__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% le_minus_one_simps(2)
tff(fact_6065_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_6066_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_6067_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_6068_le__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(4)
tff(fact_6069_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_6070_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_6071_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_6072_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_6073_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_6074_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_6075_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_6076_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_6077_add__eq__0__iff,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,Aa) ) ) ).

% add_eq_0_iff
tff(fact_6078_add__eq__0__iff,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(plus_plus_real(Aa),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,Aa) ) ) ).

% add_eq_0_iff
tff(fact_6079_add__eq__0__iff,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,Aa) ) ) ).

% add_eq_0_iff
tff(fact_6080_add__eq__0__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ).

% add_eq_0_iff
tff(fact_6081_add__eq__0__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(Aa,B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ) ) ).

% add_eq_0_iff
tff(fact_6082_ab__group__add__class_Oab__left__minus,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),Aa) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_6083_ab__group__add__class_Oab__left__minus,axiom,
    ! [Aa: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Aa)),Aa) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_6084_ab__group__add__class_Oab__left__minus,axiom,
    ! [Aa: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),Aa) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_6085_ab__group__add__class_Oab__left__minus,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Aa) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_6086_ab__group__add__class_Oab__left__minus,axiom,
    ! [Aa: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),Aa) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_6087_add_Oinverse__unique,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,Aa) = B ) ) ).

% add.inverse_unique
tff(fact_6088_add_Oinverse__unique,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(plus_plus_real(Aa),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,Aa) = B ) ) ).

% add.inverse_unique
tff(fact_6089_add_Oinverse__unique,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,Aa) = B ) ) ).

% add.inverse_unique
tff(fact_6090_add_Oinverse__unique,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,Aa) = B ) ) ).

% add.inverse_unique
tff(fact_6091_add_Oinverse__unique,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(Aa,B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) = B ) ) ).

% add.inverse_unique
tff(fact_6092_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6093_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: real,B: real] :
      ( ( Aa = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(plus_plus_real(Aa),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6094_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: complex,B: complex] :
      ( ( Aa = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6095_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(Aa),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6096_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( plus_p5714425477246183910nteger(Aa,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6097_neg__eq__iff__add__eq__0,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,Aa) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6098_neg__eq__iff__add__eq__0,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,Aa) = B )
    <=> ( aa_real_real(plus_plus_real(Aa),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6099_neg__eq__iff__add__eq__0,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,Aa) = B )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6100_neg__eq__iff__add__eq__0,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,Aa) = B )
    <=> ( aa_rat_rat(plus_plus_rat(Aa),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6101_neg__eq__iff__add__eq__0,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) = B )
    <=> ( plus_p5714425477246183910nteger(Aa,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6102_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_6103_less__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% less_minus_one_simps(2)
tff(fact_6104_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_6105_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_6106_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_6107_less__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(4)
tff(fact_6108_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_6109_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_6110_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% one_neq_neg_numeral
tff(fact_6111_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% one_neq_neg_numeral
tff(fact_6112_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_6113_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_neq_neg_numeral
tff(fact_6114_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% one_neq_neg_numeral
tff(fact_6115_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,Nb) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_6116_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_6117_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_6118_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,Nb) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_6119_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_Code_integer(numera6620942414471956472nteger,Nb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_6120_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,Xb)) = aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_6121_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(times_times_real(Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_6122_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_6123_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,Xb)) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_6124_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))) ).

% numeral_times_minus_swap
tff(fact_6125_nonzero__minus__divide__divide,axiom,
    ! [B: real,Aa: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,Aa)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(Aa),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6126_nonzero__minus__divide__divide,axiom,
    ! [B: complex,Aa: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6127_nonzero__minus__divide__divide,axiom,
    ! [B: rat,Aa: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(Aa),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6128_nonzero__minus__divide__right,axiom,
    ! [B: real,Aa: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Aa),B)) = aa_real_real(divide_divide_real(Aa),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6129_nonzero__minus__divide__right,axiom,
    ! [B: complex,Aa: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6130_nonzero__minus__divide__right,axiom,
    ! [B: rat,Aa: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),B)) = aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6131_square__eq__1__iff,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb) = one_one_int )
    <=> ( ( Xb = one_one_int )
        | ( Xb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_6132_square__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(times_times_real(Xb),Xb) = one_one_real )
    <=> ( ( Xb = one_one_real )
        | ( Xb = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_6133_square__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Xb) = one_one_complex )
    <=> ( ( Xb = one_one_complex )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_6134_square__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Xb) = one_one_rat )
    <=> ( ( Xb = one_one_rat )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_6135_square__eq__1__iff,axiom,
    ! [Xb: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xb) = one_one_Code_integer )
    <=> ( ( Xb = one_one_Code_integer )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_6136_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,Aa: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B)) ) ) ).

% group_cancel.sub2
tff(fact_6137_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,Aa: real] :
      ( ( B2 = aa_real_real(plus_plus_real(K),B) )
     => ( aa_real_real(minus_minus_real(Aa),B2) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K)),aa_real_real(minus_minus_real(Aa),B)) ) ) ).

% group_cancel.sub2
tff(fact_6138_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,Aa: complex] :
      ( ( B2 = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,K),B) )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),B2) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),B)) ) ) ).

% group_cancel.sub2
tff(fact_6139_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,Aa: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(minus_minus_rat(Aa),B2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(Aa),B)) ) ) ).

% group_cancel.sub2
tff(fact_6140_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,Aa: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(Aa,B2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(Aa,B)) ) ) ).

% group_cancel.sub2
tff(fact_6141_diff__conv__add__uminus,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_6142_diff__conv__add__uminus,axiom,
    ! [Aa: real,B: real] : aa_real_real(minus_minus_real(Aa),B) = aa_real_real(plus_plus_real(Aa),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_6143_diff__conv__add__uminus,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),B) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_6144_diff__conv__add__uminus,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(minus_minus_rat(Aa),B) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_6145_diff__conv__add__uminus,axiom,
    ! [Aa: code_integer,B: code_integer] : minus_8373710615458151222nteger(Aa,B) = plus_p5714425477246183910nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_6146_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6147_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [Aa: real,B: real] : aa_real_real(minus_minus_real(Aa),B) = aa_real_real(plus_plus_real(Aa),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6148_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),B) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6149_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(minus_minus_rat(Aa),B) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6150_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [Aa: code_integer,B: code_integer] : minus_8373710615458151222nteger(Aa,B) = plus_p5714425477246183910nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6151_dvd__neg__div,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,Aa)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)) ) ) ).

% dvd_neg_div
tff(fact_6152_dvd__neg__div,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),Aa)
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,Aa)),B) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Aa),B)) ) ) ).

% dvd_neg_div
tff(fact_6153_dvd__neg__div,axiom,
    ! [B: complex,Aa: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),Aa)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) ) ) ).

% dvd_neg_div
tff(fact_6154_dvd__neg__div,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),Aa)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,Aa)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),B)) ) ) ).

% dvd_neg_div
tff(fact_6155_dvd__neg__div,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(Aa,B)) ) ) ).

% dvd_neg_div
tff(fact_6156_dvd__div__neg,axiom,
    ! [B: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B)) ) ) ).

% dvd_div_neg
tff(fact_6157_dvd__div__neg,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),Aa)
     => ( aa_real_real(divide_divide_real(Aa),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Aa),B)) ) ) ).

% dvd_div_neg
tff(fact_6158_dvd__div__neg,axiom,
    ! [B: complex,Aa: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),Aa)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) ) ) ).

% dvd_div_neg
tff(fact_6159_dvd__div__neg,axiom,
    ! [B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),Aa)
     => ( aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),B)) ) ) ).

% dvd_div_neg
tff(fact_6160_dvd__div__neg,axiom,
    ! [B: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),Aa)
     => ( divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(Aa,B)) ) ) ).

% dvd_div_neg
tff(fact_6161_subset__Compl__self__eq,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),uminus5710092332889474511et_nat(A))
    <=> ( A = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_6162_subset__Compl__self__eq,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),uminus612125837232591019t_real(A))
    <=> ( A = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_6163_subset__Compl__self__eq,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),uminus1532241313380277803et_int(A))
    <=> ( A = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_6164_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(times_times_real(U),U))),aa_real_real(times_times_real(Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_6165_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_6166_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
     => ( ( Mb = one_one_int )
        | ( Mb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_6167_zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
    <=> ( ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( Mb = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_6168_minus__real__def,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(minus_minus_real(Xb),Y3) = aa_real_real(plus_plus_real(Xb),aa_real_real(uminus_uminus_real,Y3)) ).

% minus_real_def
tff(fact_6169_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6170_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6171_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6172_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6173_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_6174_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_6175_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_6176_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_6177_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6178_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6179_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6180_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6181_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_6182_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_6183_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_6184_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_6185_le__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(3)
tff(fact_6186_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_6187_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_6188_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_6189_le__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% le_minus_one_simps(1)
tff(fact_6190_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_6191_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_6192_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_6193_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_6194_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_6195_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_6196_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_6197_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_6198_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_6199_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_6200_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_6201_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6202_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6203_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6204_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6205_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_6206_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_6207_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_6208_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_6209_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_6210_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_6211_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_6212_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_6213_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).

% neg_one_le_numeral
tff(fact_6214_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,Mb)) ).

% neg_one_le_numeral
tff(fact_6215_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Mb)) ).

% neg_one_le_numeral
tff(fact_6216_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_le_numeral
tff(fact_6217_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_le_one
tff(fact_6218_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_6219_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_6220_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_le_one
tff(fact_6221_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_less_one
tff(fact_6222_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_less_one
tff(fact_6223_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_6224_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_6225_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_less_numeral
tff(fact_6226_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).

% neg_one_less_numeral
tff(fact_6227_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Mb)) ).

% neg_one_less_numeral
tff(fact_6228_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,Mb)) ).

% neg_one_less_numeral
tff(fact_6229_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_6230_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_6231_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_6232_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_6233_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6234_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6235_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6236_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6237_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6238_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6239_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6240_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6241_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_6242_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_6243_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_6244_uminus__numeral__One,axiom,
    aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% uminus_numeral_One
tff(fact_6245_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_6246_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(1)
tff(fact_6247_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_6248_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(1)
tff(fact_6249_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_6250_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_6251_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(2)
tff(fact_6252_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(times_times_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_6253_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(2)
tff(fact_6254_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_6255_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_6256_divide__eq__minus__1__iff,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(divide_divide_real(Aa),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( Aa = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6257_divide__eq__minus__1__iff,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( Aa = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6258_divide__eq__minus__1__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Aa),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( Aa = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6259_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,Aa: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Aa),B)) )
      <=> ( aa_real_real(times_times_real(C),B) = aa_real_real(uminus_uminus_real,Aa) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6260_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,Aa: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) )
      <=> ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) = aa_complex_complex(uminus1482373934393186551omplex,Aa) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6261_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6262_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,Aa: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Aa),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,Aa) = aa_real_real(times_times_real(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6263_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,Aa: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,Aa) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6264_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,Aa: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,Aa) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6265_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,Aa: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)) = Aa )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(times_times_real(Aa),C),Aa = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_6266_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,Aa: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) = Aa )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C),Aa = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_6267_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = Aa )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(Aa),C),Aa = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_6268_eq__minus__divide__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( ( Aa = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(Aa),C) = aa_real_real(uminus_uminus_real,B),Aa = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_6269_eq__minus__divide__eq,axiom,
    ! [Aa: complex,B: complex,C: complex] :
      ( ( Aa = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),C) = aa_complex_complex(uminus1482373934393186551omplex,B),Aa = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_6270_eq__minus__divide__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( ( Aa = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(Aa),C) = aa_rat_rat(uminus_uminus_rat,B),Aa = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_6271_power__minus,axiom,
    ! [Aa: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Aa)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ).

% power_minus
tff(fact_6272_power__minus,axiom,
    ! [Aa: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Aa)),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ).

% power_minus
tff(fact_6273_power__minus,axiom,
    ! [Aa: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),Nb) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb)) ).

% power_minus
tff(fact_6274_power__minus,axiom,
    ! [Aa: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(Aa),Nb)) ).

% power_minus
tff(fact_6275_power__minus,axiom,
    ! [Aa: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb)) ).

% power_minus
tff(fact_6276_power__minus__Bit0,axiom,
    ! [Xb: int,K: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6277_power__minus__Bit0,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6278_power__minus__Bit0,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6279_power__minus__Bit0,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6280_power__minus__Bit0,axiom,
    ! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6281_power__minus__Bit1,axiom,
    ! [Xb: int,K: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6282_power__minus__Bit1,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6283_power__minus__Bit1,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6284_power__minus__Bit1,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6285_power__minus__Bit1,axiom,
    ! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6286_real__add__less__0__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Xb),Y3)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_less_0_iff
tff(fact_6287_real__0__less__add__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(Xb),Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xb)),Y3) ) ).

% real_0_less_add_iff
tff(fact_6288_real__add__le__0__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Xb),Y3)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_le_0_iff
tff(fact_6289_real__0__le__add__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(Xb),Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),Y3) ) ).

% real_0_le_add_iff
tff(fact_6290_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_6291_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,Aa: 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(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(Aa),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6292_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6293_pos__less__minus__divide__eq,axiom,
    ! [C: real,Aa: 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),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6294_pos__less__minus__divide__eq,axiom,
    ! [C: rat,Aa: 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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6295_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,Aa: 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(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Aa),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6296_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6297_neg__less__minus__divide__eq,axiom,
    ! [C: real,Aa: 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),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(Aa),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6298_neg__less__minus__divide__eq,axiom,
    ! [C: rat,Aa: 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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6299_minus__divide__less__eq,axiom,
    ! [B: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),Aa)
    <=> $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(uminus_uminus_real,B)),aa_real_real(times_times_real(Aa),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(times_times_real(Aa),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)) ) ) ).

% minus_divide_less_eq
tff(fact_6300_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),Aa)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(Aa),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(Aa),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)) ) ) ).

% minus_divide_less_eq
tff(fact_6301_less__minus__divide__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(uminus_uminus_real,aa_real_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(times_times_real(Aa),C)),aa_real_real(uminus_uminus_real,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,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(Aa),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_6302_less__minus__divide__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(Aa),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_6303_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C) = B,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6304_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6305_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6306_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6307_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6308_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6309_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xb),Z))),Y3) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(Y3),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6310_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z))),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6311_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Y3) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Y3),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6312_add__divide__eq__if__simps_I3_J,axiom,
    ! [Aa: real,Z: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Aa),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Aa)),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6313_add__divide__eq__if__simps_I3_J,axiom,
    ! [Aa: complex,Z: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),Z))),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6314_add__divide__eq__if__simps_I3_J,axiom,
    ! [Aa: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6315_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(nat_set_decode(Mb)),nat_set_decode(Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_6316_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xb),Z))),Y3) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(Y3),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6317_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z))),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6318_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Y3) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Y3),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6319_add__divide__eq__if__simps_I5_J,axiom,
    ! [Aa: real,Z: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Aa),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Aa),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6320_add__divide__eq__if__simps_I5_J,axiom,
    ! [Aa: complex,Z: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),Z)),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6321_add__divide__eq__if__simps_I5_J,axiom,
    ! [Aa: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Aa),Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6322_add__divide__eq__if__simps_I6_J,axiom,
    ! [Aa: real,Z: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Aa),Z))),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Aa)),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6323_add__divide__eq__if__simps_I6_J,axiom,
    ! [Aa: complex,Z: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),Z))),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6324_add__divide__eq__if__simps_I6_J,axiom,
    ! [Aa: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),Z))),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6325_even__minus,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa) ) ).

% even_minus
tff(fact_6326_even__minus,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa) ) ).

% even_minus
tff(fact_6327_power2__eq__iff,axiom,
    ! [Xb: int,Y3: 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y3 )
        | ( Xb = aa_int_int(uminus_uminus_int,Y3) ) ) ) ).

% power2_eq_iff
tff(fact_6328_power2__eq__iff,axiom,
    ! [Xb: real,Y3: 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y3 )
        | ( Xb = aa_real_real(uminus_uminus_real,Y3) ) ) ) ).

% power2_eq_iff
tff(fact_6329_power2__eq__iff,axiom,
    ! [Xb: complex,Y3: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y3 )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,Y3) ) ) ) ).

% power2_eq_iff
tff(fact_6330_power2__eq__iff,axiom,
    ! [Xb: rat,Y3: 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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y3 )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,Y3) ) ) ) ).

% power2_eq_iff
tff(fact_6331_power2__eq__iff,axiom,
    ! [Xb: code_integer,Y3: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y3 )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y3) ) ) ) ).

% power2_eq_iff
tff(fact_6332_verit__less__mono__div__int2,axiom,
    ! [A: int,B2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( 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,B2),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_6333_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_6334_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,Aa: 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(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(Aa),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6335_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6336_pos__le__minus__divide__eq,axiom,
    ! [C: real,Aa: 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),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6337_pos__le__minus__divide__eq,axiom,
    ! [C: rat,Aa: 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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6338_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,Aa: 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(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Aa),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6339_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6340_neg__le__minus__divide__eq,axiom,
    ! [C: real,Aa: 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),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(Aa),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6341_neg__le__minus__divide__eq,axiom,
    ! [C: rat,Aa: 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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6342_minus__divide__le__eq,axiom,
    ! [B: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),Aa)
    <=> $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(uminus_uminus_real,B)),aa_real_real(times_times_real(Aa),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(times_times_real(Aa),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)) ) ) ).

% minus_divide_le_eq
tff(fact_6343_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),Aa)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(Aa),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(Aa),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)) ) ) ).

% minus_divide_le_eq
tff(fact_6344_le__minus__divide__eq,axiom,
    ! [Aa: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(uminus_uminus_real,aa_real_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(times_times_real(Aa),C)),aa_real_real(uminus_uminus_real,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,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(Aa),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_6345_le__minus__divide__eq,axiom,
    ! [Aa: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(Aa),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_6346_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6347_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6348_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(uminus_uminus_real,aa_num_real(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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6349_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6350_power2__eq__1__iff,axiom,
    ! [Aa: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( Aa = one_one_int )
        | ( Aa = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_6351_power2__eq__1__iff,axiom,
    ! [Aa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( Aa = one_one_real )
        | ( Aa = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_6352_power2__eq__1__iff,axiom,
    ! [Aa: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( Aa = one_one_complex )
        | ( Aa = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_6353_power2__eq__1__iff,axiom,
    ! [Aa: rat] :
      ( ( aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( Aa = one_one_rat )
        | ( Aa = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_6354_power2__eq__1__iff,axiom,
    ! [Aa: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( Aa = one_one_Code_integer )
        | ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_6355_uminus__power__if,axiom,
    ! [Aa: int,Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Aa)),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_int(aa_int_fun_nat_int(power_power_int,Aa),Nb),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb))) ).

% uminus_power_if
tff(fact_6356_uminus__power__if,axiom,
    ! [Aa: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Aa)),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_real(aa_real_fun_nat_real(power_power_real,Aa),Nb),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb))) ).

% uminus_power_if
tff(fact_6357_uminus__power__if,axiom,
    ! [Aa: complex,Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),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_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),Nb))) ).

% uminus_power_if
tff(fact_6358_uminus__power__if,axiom,
    ! [Aa: rat,Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),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_rat(power_power_rat(Aa),Nb),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Aa),Nb))) ).

% uminus_power_if
tff(fact_6359_uminus__power__if,axiom,
    ! [Aa: code_integer,Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),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_Code_integer(power_8256067586552552935nteger(Aa),Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb))) ).

% uminus_power_if
tff(fact_6360_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6361_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6362_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6363_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6364_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6365_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_6366_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_6367_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_6368_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_6369_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_6370_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_6371_zdiv__zminus2__eq__if,axiom,
    ! [B: int,Aa: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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,Aa),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_6372_zdiv__zminus1__eq__if,axiom,
    ! [B: int,Aa: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,Aa)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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,Aa),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_6373_zminus1__lemma,axiom,
    ! [Aa: int,B: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(Aa,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,Aa),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_6374_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6375_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6376_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(uminus_uminus_real,aa_num_real(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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6377_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6378_square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_6379_square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_6380_square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_6381_square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_6382_minus__power__mult__self,axiom,
    ! [Aa: 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,aa_int_int(uminus_uminus_int,Aa)),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Aa)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6383_minus__power__mult__self,axiom,
    ! [Aa: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Aa)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Aa)),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6384_minus__power__mult__self,axiom,
    ! [Aa: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Aa)),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6385_minus__power__mult__self,axiom,
    ! [Aa: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Nb)) = aa_nat_rat(power_power_rat(Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6386_minus__power__mult__self,axiom,
    ! [Aa: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6387_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),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),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_6388_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),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),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_6389_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),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),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_6390_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),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),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_6391_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),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),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_6392_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_6393_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_6394_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_6395_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_6396_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Mb),Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% divmod_nat_def
tff(fact_6397_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_6398_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_6399_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_6400_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_6401_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_6402_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_6403_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_6404_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_6405_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_hf,Xb)) ).

% set_decode_def
tff(fact_6406_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_6407_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_6408_compl__le__compl__iff,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xb)),uminus1532241313380277803et_int(Y3))
    <=> aa_set_int_o(ord_less_eq_set_int(Y3),Xb) ) ).

% compl_le_compl_iff
tff(fact_6409_minus__numeral__div__numeral,axiom,
    ! [Mb: 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,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(Mb,Nb))) ).

% minus_numeral_div_numeral
tff(fact_6410_numeral__div__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(Mb,Nb))) ).

% numeral_div_minus_numeral
tff(fact_6411_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_hg((P)),Q)),X3)
    <=> ( (P)
        & aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Q),X3) ) ) ).

% split_part
tff(fact_6412_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_hh((P)),Q)),X3)
    <=> ( (P)
        & aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q),X3) ) ) ).

% split_part
tff(fact_6413_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aTP_Lamm_hi),Prod) ).

% prod.disc_eq_case
tff(fact_6414_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_hj),Prod) ).

% prod.disc_eq_case
tff(fact_6415_Collect__case__prod__mono,axiom,
    ! [A: fun_nat_fun_nat_o,B2: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(A,B2)
     => ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(A)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(B2))) ) ).

% Collect_case_prod_mono
tff(fact_6416_Collect__case__prod__mono,axiom,
    ! [A: fun_int_fun_int_o,B2: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A,B2)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(A)),collec213857154873943460nt_int(produc4947309494688390418_int_o(B2))) ) ).

% Collect_case_prod_mono
tff(fact_6417_Compl__eq,axiom,
    ! [A: set_real] : uminus612125837232591019t_real(A) = collect_real(aTP_Lamm_hk(A)) ).

% Compl_eq
tff(fact_6418_Compl__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A) = collec213857154873943460nt_int(aTP_Lamm_hl(A)) ).

% Compl_eq
tff(fact_6419_Compl__eq,axiom,
    ! [A: set_complex] : uminus8566677241136511917omplex(A) = collect_complex(aTP_Lamm_hm(A)) ).

% Compl_eq
tff(fact_6420_Compl__eq,axiom,
    ! [A: set_set_nat] : uminus613421341184616069et_nat(A) = collect_set_nat(aTP_Lamm_hn(A)) ).

% Compl_eq
tff(fact_6421_Compl__eq,axiom,
    ! [A: set_nat] : uminus5710092332889474511et_nat(A) = collect_nat(aTP_Lamm_ho(A)) ).

% Compl_eq
tff(fact_6422_Compl__eq,axiom,
    ! [A: set_int] : uminus1532241313380277803et_int(A) = collect_int(aTP_Lamm_hp(A)) ).

% Compl_eq
tff(fact_6423_Collect__neg__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aTP_Lamm_hq(P)) = uminus6221592323253981072nt_int(collec213857154873943460nt_int(P)) ).

% Collect_neg_eq
tff(fact_6424_Collect__neg__eq,axiom,
    ! [P: fun_complex_o] : collect_complex(aTP_Lamm_hr(P)) = uminus8566677241136511917omplex(collect_complex(P)) ).

% Collect_neg_eq
tff(fact_6425_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_o] : collect_set_nat(aTP_Lamm_hs(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_6426_Collect__neg__eq,axiom,
    ! [P: fun_nat_o] : collect_nat(aTP_Lamm_ht(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_6427_Collect__neg__eq,axiom,
    ! [P: fun_int_o] : collect_int(aTP_Lamm_hu(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_6428_uminus__set__def,axiom,
    ! [A: set_real] : uminus612125837232591019t_real(A) = collect_real(uminus2572480474538279996real_o(aTP_Lamm_a(A))) ).

% uminus_set_def
tff(fact_6429_uminus__set__def,axiom,
    ! [A: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A) = collec213857154873943460nt_int(uminus103585356917985579_int_o(aTP_Lamm_aa(A))) ).

% uminus_set_def
tff(fact_6430_uminus__set__def,axiom,
    ! [A: set_complex] : uminus8566677241136511917omplex(A) = collect_complex(uminus4035047602201070266plex_o(aTP_Lamm_ab(A))) ).

% uminus_set_def
tff(fact_6431_uminus__set__def,axiom,
    ! [A: set_set_nat] : uminus613421341184616069et_nat(A) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_ac(A))) ).

% uminus_set_def
tff(fact_6432_uminus__set__def,axiom,
    ! [A: set_nat] : uminus5710092332889474511et_nat(A) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ad(A))) ).

% uminus_set_def
tff(fact_6433_uminus__set__def,axiom,
    ! [A: set_int] : uminus1532241313380277803et_int(A) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_ae(A))) ).

% uminus_set_def
tff(fact_6434_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_hv),Qr) ).

% Divides.adjust_div_def
tff(fact_6435_compl__le__swap2,axiom,
    ! [Y3: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Y3)),Xb)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xb)),Y3) ) ).

% compl_le_swap2
tff(fact_6436_compl__le__swap1,axiom,
    ! [Y3: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y3),uminus1532241313380277803et_int(Xb))
     => aa_set_int_o(ord_less_eq_set_int(Xb),uminus1532241313380277803et_int(Y3)) ) ).

% compl_le_swap1
tff(fact_6437_compl__mono,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y3)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Y3)),uminus1532241313380277803et_int(Xb)) ) ).

% compl_mono
tff(fact_6438_diff__shunt__var,axiom,
    ! [Xb: set_real,Y3: set_real] :
      ( ( minus_minus_set_real(Xb,Y3) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xb),Y3) ) ).

% diff_shunt_var
tff(fact_6439_diff__shunt__var,axiom,
    ! [Xb: set_nat,Y3: set_nat] :
      ( ( minus_minus_set_nat(Xb,Y3) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y3) ) ).

% diff_shunt_var
tff(fact_6440_diff__shunt__var,axiom,
    ! [Xb: set_int,Y3: set_int] :
      ( ( minus_minus_set_int(Xb,Y3) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xb),Y3) ) ).

% diff_shunt_var
tff(fact_6441_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6442_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6443_dbl__dec__simps_I4_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6444_dbl__dec__simps_I4_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6445_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6446_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_6447_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y3 )
     => ( Y3 = $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_6448_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_6449_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(
        K = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: int,
            l:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,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_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_6450_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(
        K = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: real,
            l:= aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,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_real_real(plus_plus_real(l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_6451_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_complex(ring_17405671764205052669omplex,K) = $ite(
        K = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: complex,
            l:= aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,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_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_6452_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(
        K = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: rat,
            l:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,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_rat_rat(plus_plus_rat(l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_6453_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K) = $ite(
        K = zero_zero_int,
        zero_z3403309356797280102nteger,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))),
          $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(ring_18347121197199848620nteger,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,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_6454_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_hw(D))) ).

% int_ge_less_than_def
tff(fact_6455_and_Oright__idem,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),B)),B) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),B) ).

% and.right_idem
tff(fact_6456_and_Oright__idem,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),B) ).

% and.right_idem
tff(fact_6457_and_Oleft__idem,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),B) ).

% and.left_idem
tff(fact_6458_and_Oleft__idem,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),B) ).

% and.left_idem
tff(fact_6459_and_Oidem,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),Aa) = Aa ).

% and.idem
tff(fact_6460_and_Oidem,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),Aa) = Aa ).

% and.idem
tff(fact_6461_bit_Oconj__zero__right,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_6462_bit_Oconj__zero__left,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),Xb) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_6463_zero__and__eq,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),Aa) = zero_zero_int ).

% zero_and_eq
tff(fact_6464_zero__and__eq,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% zero_and_eq
tff(fact_6465_and__zero__eq,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_6466_and__zero__eq,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_6467_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_6468_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_6469_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_6470_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_6471_bit_Oconj__one__right,axiom,
    ! [Xb: code_integer] : bit_se3949692690581998587nteger(Xb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = Xb ).

% bit.conj_one_right
tff(fact_6472_bit_Oconj__one__right,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),aa_int_int(uminus_uminus_int,one_one_int)) = Xb ).

% bit.conj_one_right
tff(fact_6473_and_Oright__neutral,axiom,
    ! [Aa: code_integer] : bit_se3949692690581998587nteger(Aa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = Aa ).

% and.right_neutral
tff(fact_6474_and_Oright__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),aa_int_int(uminus_uminus_int,one_one_int)) = Aa ).

% and.right_neutral
tff(fact_6475_and_Oleft__neutral,axiom,
    ! [Aa: code_integer] : bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Aa) = Aa ).

% and.left_neutral
tff(fact_6476_and_Oleft__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,one_one_int)),Aa) = Aa ).

% and.left_neutral
tff(fact_6477_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6478_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6479_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6480_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_6481_of__int__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).

% of_int_numeral
tff(fact_6482_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).

% of_int_numeral
tff(fact_6483_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).

% of_int_numeral
tff(fact_6484_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_6485_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_6486_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_6487_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_6488_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_6489_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_6490_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_6491_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_6492_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = one_one_complex )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_6493_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_6494_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_6495_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_6496_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_6497_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_6498_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_6499_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_6500_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(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_6501_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_6502_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_6503_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_6504_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(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_6505_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_6506_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_6507_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_6508_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_6509_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Xb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,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_6510_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_6511_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_6512_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_6513_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,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_6514_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_6515_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_6516_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_6517_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,Z)),Nb) ).

% of_int_power
tff(fact_6518_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_6519_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_6520_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_6521_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_6522_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_6523_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_6524_and__numerals_I2_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = one_one_int ).

% and_numerals(2)
tff(fact_6525_and__numerals_I2_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = one_one_nat ).

% and_numerals(2)
tff(fact_6526_and__numerals_I8_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_6527_and__numerals_I8_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))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_6528_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( 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,Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ) ).

% ln_le_cancel_iff
tff(fact_6529_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_6530_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_6531_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_6532_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_6533_dbl__dec__simps_I2_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,zero_zero_real) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_dec_simps(2)
tff(fact_6534_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_6535_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_6536_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_6537_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_dec_simps(1)
tff(fact_6538_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_dec_simps(1)
tff(fact_6539_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K))) ).

% dbl_dec_simps(1)
tff(fact_6540_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_dec_simps(1)
tff(fact_6541_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_dec_simps(1)
tff(fact_6542_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_inc_simps(1)
tff(fact_6543_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_inc_simps(1)
tff(fact_6544_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K))) ).

% dbl_inc_simps(1)
tff(fact_6545_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_inc_simps(1)
tff(fact_6546_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_inc_simps(1)
tff(fact_6547_and__numerals_I1_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = zero_zero_int ).

% and_numerals(1)
tff(fact_6548_and__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_6549_and__numerals_I5_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_6550_and__numerals_I5_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))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_6551_and__numerals_I3_J,axiom,
    ! [Xb: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = 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,Xb)),aa_num_int(numeral_numeral_int,Y3))) ).

% and_numerals(3)
tff(fact_6552_and__numerals_I3_J,axiom,
    ! [Xb: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = 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_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y3))) ).

% and_numerals(3)
tff(fact_6553_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_6554_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_6555_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_6556_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_6557_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_6558_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_6559_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_6560_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_6561_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_6562_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_6563_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_6564_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_6565_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6566_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6567_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_6568_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_6569_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_6570_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_6571_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6572_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6573_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_6574_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_6575_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_6576_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_6577_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_6578_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_6579_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_6580_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_6581_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_6582_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_6583_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_6584_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_6585_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_6586_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_6587_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_6588_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_6589_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_6590_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_6591_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y3) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Y3 = 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_6592_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y3) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
    <=> ( Y3 = 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_6593_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y3) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) )
    <=> ( Y3 = 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_6594_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Y3 = 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_6595_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6596_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_int_real(ring_1_of_int_real,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6597_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) = aa_int_rat(ring_1_of_int_rat,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6598_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: 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,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6599_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_6600_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_6601_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_6602_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_6603_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_6604_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_6605_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_6606_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_6607_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_6608_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_6609_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_6610_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_6611_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_6612_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_6613_and__numerals_I4_J,axiom,
    ! [Xb: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = 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,Xb)),aa_num_int(numeral_numeral_int,Y3))) ).

% and_numerals(4)
tff(fact_6614_and__numerals_I4_J,axiom,
    ! [Xb: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = 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_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y3))) ).

% and_numerals(4)
tff(fact_6615_and__numerals_I6_J,axiom,
    ! [Xb: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = 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,Xb)),aa_num_int(numeral_numeral_int,Y3))) ).

% and_numerals(6)
tff(fact_6616_and__numerals_I6_J,axiom,
    ! [Xb: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = 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_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y3))) ).

% and_numerals(6)
tff(fact_6617_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_6618_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_6619_of__int__le__numeral__power__cancel__iff,axiom,
    ! [Aa: 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,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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_6620_of__int__le__numeral__power__cancel__iff,axiom,
    ! [Aa: 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,Aa)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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_6621_of__int__le__numeral__power__cancel__iff,axiom,
    ! [Aa: 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,Aa)),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,Aa),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_6622_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),aa_int_real(ring_1_of_int_real,Aa))
    <=> 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) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6623_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,Aa))
    <=> 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) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6624_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: 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,Aa))
    <=> 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) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6625_of__int__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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_6626_of__int__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Aa)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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_6627_of__int__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Aa)),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,Aa),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_6628_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),aa_int_real(ring_1_of_int_real,Aa))
    <=> 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) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6629_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,Aa))
    <=> 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) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6630_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: 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,Aa))
    <=> 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) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6631_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = aa_int_int(ring_1_of_int_int,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6632_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb) = aa_int_real(ring_1_of_int_real,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6633_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6634_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) = aa_int_rat(ring_1_of_int_rat,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6635_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Nb) = aa_int_Code_integer(ring_18347121197199848620nteger,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6636_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6637_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y3) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6638_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y3) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6639_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y3) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6640_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y3) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Nb) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6641_and__numerals_I7_J,axiom,
    ! [Xb: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = 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,Xb)),aa_num_int(numeral_numeral_int,Y3)))) ).

% and_numerals(7)
tff(fact_6642_and__numerals_I7_J,axiom,
    ! [Xb: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% and_numerals(7)
tff(fact_6643_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),aa_int_real(ring_1_of_int_real,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),Aa) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6644_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),Aa) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6645_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),Aa) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6646_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),Aa) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6647_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: 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,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6648_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Aa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6649_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: 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,Aa)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6650_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: 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,Aa)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6651_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),Aa) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6652_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),aa_int_real(ring_1_of_int_real,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),Aa) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6653_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),Aa) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6654_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),Aa) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6655_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Aa)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6656_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6657_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Aa)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6658_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Aa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6659_and_Oleft__commute,axiom,
    ! [B: int,Aa: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),C)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),C)) ).

% and.left_commute
tff(fact_6660_and_Oleft__commute,axiom,
    ! [B: nat,Aa: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.left_commute
tff(fact_6661_and_Ocommute,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),B) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),Aa) ).

% and.commute
tff(fact_6662_and_Ocommute,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),Aa) ).

% and.commute
tff(fact_6663_and_Oassoc,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),C)) ).

% and.assoc
tff(fact_6664_and_Oassoc,axiom,
    ! [Aa: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.assoc
tff(fact_6665_of__int__and__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_and_eq
tff(fact_6666_mult__of__int__commute,axiom,
    ! [Xb: int,Y3: real] : aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Xb)),Y3) = aa_real_real(times_times_real(Y3),aa_int_real(ring_1_of_int_real,Xb)) ).

% mult_of_int_commute
tff(fact_6667_mult__of__int__commute,axiom,
    ! [Xb: int,Y3: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,Xb)),Y3) = aa_rat_rat(times_times_rat(Y3),aa_int_rat(ring_1_of_int_rat,Xb)) ).

% mult_of_int_commute
tff(fact_6668_mult__of__int__commute,axiom,
    ! [Xb: int,Y3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,Xb)),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),aa_int_int(ring_1_of_int_int,Xb)) ).

% mult_of_int_commute
tff(fact_6669_of__int__max,axiom,
    ! [Xb: int,Y3: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y3)) = aa_real_real(ord_max_real(aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,Y3)) ).

% of_int_max
tff(fact_6670_of__int__max,axiom,
    ! [Xb: int,Y3: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y3)) = aa_rat_rat(ord_max_rat(aa_int_rat(ring_1_of_int_rat,Xb)),aa_int_rat(ring_1_of_int_rat,Y3)) ).

% of_int_max
tff(fact_6671_of__int__max,axiom,
    ! [Xb: int,Y3: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,Xb)),aa_int_int(ring_1_of_int_int,Y3)) ).

% of_int_max
tff(fact_6672_of__int__max,axiom,
    ! [Xb: int,Y3: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y3)) = ord_max_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Xb),aa_int_Code_integer(ring_18347121197199848620nteger,Y3)) ).

% of_int_max
tff(fact_6673_and__eq__minus__1__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( bit_se3949692690581998587nteger(Aa,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
        & ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_6674_and__eq__minus__1__iff,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( Aa = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_6675_AND__upper2_H,axiom,
    ! [Y3: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),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),Y3)),Z) ) ) ).

% AND_upper2'
tff(fact_6676_AND__upper1_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y3),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_6677_AND__upper2,axiom,
    ! [Y3: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y3)),Y3) ) ).

% AND_upper2
tff(fact_6678_AND__upper1,axiom,
    ! [Xb: int,Y3: 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),Y3)),Xb) ) ).

% AND_upper1
tff(fact_6679_AND__lower,axiom,
    ! [Xb: int,Y3: 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),Y3)) ) ).

% AND_lower
tff(fact_6680_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_6681_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_6682_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_6683_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_6684_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_6685_AND__upper2_H_H,axiom,
    ! [Y3: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y3)),Z) ) ) ).

% AND_upper2''
tff(fact_6686_AND__upper1_H_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y3),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_6687_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_6688_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(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_6689_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(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_6690_even__and__iff,axiom,
    ! [Aa: 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))),bit_se3949692690581998587nteger(Aa,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6691_even__and__iff,axiom,
    ! [Aa: 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(bit_se725231765392027082nd_int,Aa),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6692_even__and__iff,axiom,
    ! [Aa: 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(bit_se727722235901077358nd_nat,Aa),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6693_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_6694_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(plus_plus_real(one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_6695_ln__mult,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_real(ln_ln_real,aa_real_real(times_times_real(Xb),Y3)) = aa_real_real(plus_plus_real(aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y3)) ) ) ) ).

% ln_mult
tff(fact_6696_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(minus_minus_real(Xb),one_one_real) )
       => ( Xb = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_6697_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_6698_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_6699_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_6700_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_6701_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_6702_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_6703_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_6704_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_6705_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K)) ).

% of_int_neg_numeral
tff(fact_6706_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K)) ).

% of_int_neg_numeral
tff(fact_6707_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) ).

% of_int_neg_numeral
tff(fact_6708_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K)) ).

% of_int_neg_numeral
tff(fact_6709_int__le__real__less,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Mb)),one_one_real)) ) ).

% int_le_real_less
tff(fact_6710_int__less__real__le,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,Mb)) ) ).

% int_less_real_le
tff(fact_6711_real__of__int__div__aux,axiom,
    ! [Xb: int,D: int] : aa_real_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(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(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_6712_one__and__eq,axiom,
    ! [Aa: code_integer] : bit_se3949692690581998587nteger(one_one_Code_integer,Aa) = modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6713_one__and__eq,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),Aa) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6714_one__and__eq,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6715_and__one__eq,axiom,
    ! [Aa: code_integer] : bit_se3949692690581998587nteger(Aa,one_one_Code_integer) = modulo364778990260209775nteger(Aa,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6716_and__one__eq,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),one_one_int) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6717_and__one__eq,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Aa),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6718_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_6719_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(minus_minus_real(Xb),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_6720_ln__diff__le,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y3))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xb),Y3)),Y3)) ) ) ).

% ln_diff_le
tff(fact_6721_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(plus_plus_real(one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_6722_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(minus_minus_real(aa_real_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_6723_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_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_6724_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(minus_minus_real(one_one_real),Xb))),aa_real_real(uminus_uminus_real,Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_6725_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_6726_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_6727_dbl__dec__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,Xb) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Xb)),one_one_complex) ).

% dbl_dec_def
tff(fact_6728_dbl__dec__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu6075765906172075777c_real,Xb) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Xb),Xb)),one_one_real) ).

% dbl_dec_def
tff(fact_6729_dbl__dec__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,Xb) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).

% dbl_dec_def
tff(fact_6730_dbl__dec__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_dec_def
tff(fact_6731_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_6732_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(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(plus_plus_real(one_one_real),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6733_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_6734_artanh__def,axiom,
    ! [Xb: real] : aa_real_real(artanh_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(one_one_real),Xb)),aa_real_real(minus_minus_real(one_one_real),Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_6735_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)))
      & ! [Y2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y2)),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,Y2),one_one_int))) )
         => ( Y2 = X2 ) ) ) ).

% floor_exists1
tff(fact_6736_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)))
      & ! [Y2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y2)),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,Y2),one_one_int))) )
         => ( Y2 = X2 ) ) ) ).

% floor_exists1
tff(fact_6737_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_6738_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_6739_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_hx(D))) ).

% int_ge_less_than2_def
tff(fact_6740_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),Xb))),Xb))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_6741_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(divide_divide_real(aa_real_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(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_6742_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% abs_numeral
tff(fact_6743_abs__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_numeral
tff(fact_6744_abs__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_numeral
tff(fact_6745_abs__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_numeral
tff(fact_6746_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_6747_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_6748_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_6749_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_6750_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_6751_abs__mult__self__eq,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Aa)),abs_abs_Code_integer(Aa)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Aa) ).

% abs_mult_self_eq
tff(fact_6752_abs__mult__self__eq,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Aa)) = aa_real_real(times_times_real(Aa),Aa) ).

% abs_mult_self_eq
tff(fact_6753_abs__mult__self__eq,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Aa)) = aa_rat_rat(times_times_rat(Aa),Aa) ).

% abs_mult_self_eq
tff(fact_6754_abs__mult__self__eq,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,Aa)) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Aa) ).

% abs_mult_self_eq
tff(fact_6755_abs__add__abs,axiom,
    ! [Aa: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(Aa),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(Aa),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_6756_abs__add__abs,axiom,
    ! [Aa: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B))) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_6757_abs__add__abs,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_6758_abs__add__abs,axiom,
    ! [Aa: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_6759_abs__divide,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(abs_abs_complex,Aa)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_divide
tff(fact_6760_abs__divide,axiom,
    ! [Aa: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Aa),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_6761_abs__divide,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Aa),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_divide
tff(fact_6762_tanh__real__le__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ).

% tanh_real_le_iff
tff(fact_6763_abs__of__nonneg,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => ( abs_abs_Code_integer(Aa) = Aa ) ) ).

% abs_of_nonneg
tff(fact_6764_abs__of__nonneg,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_real(abs_abs_real,Aa) = Aa ) ) ).

% abs_of_nonneg
tff(fact_6765_abs__of__nonneg,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_rat(abs_abs_rat,Aa) = Aa ) ) ).

% abs_of_nonneg
tff(fact_6766_abs__of__nonneg,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_int(abs_abs_int,Aa) = Aa ) ) ).

% abs_of_nonneg
tff(fact_6767_abs__le__self__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Aa)),Aa)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa) ) ).

% abs_le_self_iff
tff(fact_6768_abs__le__self__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% abs_le_self_iff
tff(fact_6769_abs__le__self__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% abs_le_self_iff
tff(fact_6770_abs__le__self__iff,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),Aa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ).

% abs_le_self_iff
tff(fact_6771_abs__le__zero__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Aa)),zero_z3403309356797280102nteger)
    <=> ( Aa = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_6772_abs__le__zero__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),zero_zero_real)
    <=> ( Aa = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_6773_abs__le__zero__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),zero_zero_rat)
    <=> ( Aa = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_6774_abs__le__zero__iff,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),zero_zero_int)
    <=> ( Aa = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_6775_zero__less__abs__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(Aa))
    <=> ( Aa != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_6776_zero__less__abs__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,Aa))
    <=> ( Aa != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_6777_zero__less__abs__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,Aa))
    <=> ( Aa != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_6778_zero__less__abs__iff,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,Aa))
    <=> ( Aa != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_6779_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_neg_numeral
tff(fact_6780_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_neg_numeral
tff(fact_6781_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_neg_numeral
tff(fact_6782_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% abs_neg_numeral
tff(fact_6783_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_6784_abs__neg__one,axiom,
    aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,one_one_real)) = one_one_real ).

% abs_neg_one
tff(fact_6785_abs__neg__one,axiom,
    aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_6786_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_6787_abs__power__minus,axiom,
    ! [Aa: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Aa)),Nb)) = aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) ).

% abs_power_minus
tff(fact_6788_abs__power__minus,axiom,
    ! [Aa: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Aa)),Nb)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) ).

% abs_power_minus
tff(fact_6789_abs__power__minus,axiom,
    ! [Aa: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Nb)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(Aa),Nb)) ).

% abs_power_minus
tff(fact_6790_abs__power__minus,axiom,
    ! [Aa: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Nb)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb)) ).

% abs_power_minus
tff(fact_6791_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_6792_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_6793_divide__le__0__abs__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Aa),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6794_divide__le__0__abs__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6795_zero__le__divide__abs__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Aa),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6796_zero__le__divide__abs__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(abs_abs_rat,B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6797_abs__of__nonpos,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
     => ( aa_real_real(abs_abs_real,Aa) = aa_real_real(uminus_uminus_real,Aa) ) ) ).

% abs_of_nonpos
tff(fact_6798_abs__of__nonpos,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(Aa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ) ) ).

% abs_of_nonpos
tff(fact_6799_abs__of__nonpos,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,Aa) = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ).

% abs_of_nonpos
tff(fact_6800_abs__of__nonpos,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_int(abs_abs_int,Aa) = aa_int_int(uminus_uminus_int,Aa) ) ) ).

% abs_of_nonpos
tff(fact_6801_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_6802_and__nat__numerals_I1_J,axiom,
    ! [Y3: 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,Y3))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_6803_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_6804_zero__less__power__abs__iff,axiom,
    ! [Aa: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(Aa)),Nb))
    <=> ( ( Aa != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6805_zero__less__power__abs__iff,axiom,
    ! [Aa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Aa)),Nb))
    <=> ( ( Aa != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6806_zero__less__power__abs__iff,axiom,
    ! [Aa: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,Aa)),Nb))
    <=> ( ( Aa != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6807_zero__less__power__abs__iff,axiom,
    ! [Aa: 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,aa_int_int(abs_abs_int,Aa)),Nb))
    <=> ( ( Aa != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6808_abs__power2,axiom,
    ! [Aa: rat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6809_abs__power2,axiom,
    ! [Aa: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6810_abs__power2,axiom,
    ! [Aa: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6811_abs__power2,axiom,
    ! [Aa: int] : aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6812_power2__abs,axiom,
    ! [Aa: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6813_power2__abs,axiom,
    ! [Aa: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6814_power2__abs,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6815_power2__abs,axiom,
    ! [Aa: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6816_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_6817_and__nat__numerals_I2_J,axiom,
    ! [Y3: 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,Y3))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_6818_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_6819_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_6820_power__even__abs__numeral,axiom,
    ! [W: num,Aa: 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_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6821_power__even__abs__numeral,axiom,
    ! [W: num,Aa: code_integer] :
      ( 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_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(Aa)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6822_power__even__abs__numeral,axiom,
    ! [W: num,Aa: 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_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Aa)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6823_power__even__abs__numeral,axiom,
    ! [W: num,Aa: 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_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,Aa)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6824_abs__le__D1,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ).

% abs_le_D1
tff(fact_6825_abs__le__D1,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Aa)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),B) ) ).

% abs_le_D1
tff(fact_6826_abs__le__D1,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B) ) ).

% abs_le_D1
tff(fact_6827_abs__le__D1,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B) ) ).

% abs_le_D1
tff(fact_6828_abs__ge__self,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(abs_abs_real,Aa)) ).

% abs_ge_self
tff(fact_6829_abs__ge__self,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),abs_abs_Code_integer(Aa)) ).

% abs_ge_self
tff(fact_6830_abs__ge__self,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(abs_abs_rat,Aa)) ).

% abs_ge_self
tff(fact_6831_abs__ge__self,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(abs_abs_int,Aa)) ).

% abs_ge_self
tff(fact_6832_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_6833_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_6834_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_6835_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_6836_abs__mult,axiom,
    ! [Aa: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Aa)),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_6837_abs__mult,axiom,
    ! [Aa: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(times_times_real(Aa),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_6838_abs__mult,axiom,
    ! [Aa: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Aa),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_6839_abs__mult,axiom,
    ! [Aa: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_6840_power__abs,axiom,
    ! [Aa: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(Aa),Nb)) = aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,Aa)),Nb) ).

% power_abs
tff(fact_6841_power__abs,axiom,
    ! [Aa: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(Aa)),Nb) ).

% power_abs
tff(fact_6842_power__abs,axiom,
    ! [Aa: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Aa)),Nb) ).

% power_abs
tff(fact_6843_power__abs,axiom,
    ! [Aa: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,Aa)),Nb) ).

% power_abs
tff(fact_6844_abs__ge__zero,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(Aa)) ).

% abs_ge_zero
tff(fact_6845_abs__ge__zero,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,Aa)) ).

% abs_ge_zero
tff(fact_6846_abs__ge__zero,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,Aa)) ).

% abs_ge_zero
tff(fact_6847_abs__ge__zero,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,Aa)) ).

% abs_ge_zero
tff(fact_6848_abs__not__less__zero,axiom,
    ! [Aa: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Aa)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_6849_abs__not__less__zero,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Aa)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_6850_abs__not__less__zero,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Aa)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_6851_abs__not__less__zero,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Aa)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_6852_abs__of__pos,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Aa)
     => ( abs_abs_Code_integer(Aa) = Aa ) ) ).

% abs_of_pos
tff(fact_6853_abs__of__pos,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_real(abs_abs_real,Aa) = Aa ) ) ).

% abs_of_pos
tff(fact_6854_abs__of__pos,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_rat(abs_abs_rat,Aa) = Aa ) ) ).

% abs_of_pos
tff(fact_6855_abs__of__pos,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_int(abs_abs_int,Aa) = Aa ) ) ).

% abs_of_pos
tff(fact_6856_abs__triangle__ineq,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(Aa,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(Aa),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_6857_abs__triangle__ineq,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(Aa),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_6858_abs__triangle__ineq,axiom,
    ! [Aa: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(Aa),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_6859_abs__triangle__ineq,axiom,
    ! [Aa: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_6860_abs__mult__less,axiom,
    ! [Aa: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Aa)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Aa)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_6861_abs__mult__less,axiom,
    ! [Aa: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Aa)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B))),aa_real_real(times_times_real(C),D)) ) ) ).

% abs_mult_less
tff(fact_6862_abs__mult__less,axiom,
    ! [Aa: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Aa)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_6863_abs__mult__less,axiom,
    ! [Aa: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Aa)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ) ).

% abs_mult_less
tff(fact_6864_abs__triangle__ineq2,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(Aa),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(Aa,B))) ).

% abs_triangle_ineq2
tff(fact_6865_abs__triangle__ineq2,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Aa),B))) ).

% abs_triangle_ineq2
tff(fact_6866_abs__triangle__ineq2,axiom,
    ! [Aa: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Aa),B))) ).

% abs_triangle_ineq2
tff(fact_6867_abs__triangle__ineq2,axiom,
    ! [Aa: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B))) ).

% abs_triangle_ineq2
tff(fact_6868_abs__triangle__ineq3,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(Aa),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(Aa,B))) ).

% abs_triangle_ineq3
tff(fact_6869_abs__triangle__ineq3,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Aa),B))) ).

% abs_triangle_ineq3
tff(fact_6870_abs__triangle__ineq3,axiom,
    ! [Aa: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Aa),B))) ).

% abs_triangle_ineq3
tff(fact_6871_abs__triangle__ineq3,axiom,
    ! [Aa: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B))) ).

% abs_triangle_ineq3
tff(fact_6872_abs__triangle__ineq2__sym,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(Aa),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,Aa))) ).

% abs_triangle_ineq2_sym
tff(fact_6873_abs__triangle__ineq2__sym,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),Aa))) ).

% abs_triangle_ineq2_sym
tff(fact_6874_abs__triangle__ineq2__sym,axiom,
    ! [Aa: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),Aa))) ).

% abs_triangle_ineq2_sym
tff(fact_6875_abs__triangle__ineq2__sym,axiom,
    ! [Aa: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Aa))) ).

% abs_triangle_ineq2_sym
tff(fact_6876_nonzero__abs__divide,axiom,
    ! [B: real,Aa: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Aa),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_6877_nonzero__abs__divide,axiom,
    ! [B: rat,Aa: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Aa),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_6878_abs__ge__minus__self,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),aa_real_real(abs_abs_real,Aa)) ).

% abs_ge_minus_self
tff(fact_6879_abs__ge__minus__self,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),abs_abs_Code_integer(Aa)) ).

% abs_ge_minus_self
tff(fact_6880_abs__ge__minus__self,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(abs_abs_rat,Aa)) ).

% abs_ge_minus_self
tff(fact_6881_abs__ge__minus__self,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(abs_abs_int,Aa)) ).

% abs_ge_minus_self
tff(fact_6882_abs__le__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),B) ) ) ).

% abs_le_iff
tff(fact_6883_abs__le__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Aa)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),B) ) ) ).

% abs_le_iff
tff(fact_6884_abs__le__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),B) ) ) ).

% abs_le_iff
tff(fact_6885_abs__le__iff,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),B) ) ) ).

% abs_le_iff
tff(fact_6886_abs__le__D2,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),B) ) ).

% abs_le_D2
tff(fact_6887_abs__le__D2,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Aa)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),B) ) ).

% abs_le_D2
tff(fact_6888_abs__le__D2,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),B) ) ).

% abs_le_D2
tff(fact_6889_abs__le__D2,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),B) ) ).

% abs_le_D2
tff(fact_6890_abs__leI,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),B) ) ) ).

% abs_leI
tff(fact_6891_abs__leI,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Aa)),B) ) ) ).

% abs_leI
tff(fact_6892_abs__leI,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),B) ) ) ).

% abs_leI
tff(fact_6893_abs__leI,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),B) ) ) ).

% abs_leI
tff(fact_6894_abs__less__iff,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Aa)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Aa)),B) ) ) ).

% abs_less_iff
tff(fact_6895_abs__less__iff,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Aa)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Aa)),B) ) ) ).

% abs_less_iff
tff(fact_6896_abs__less__iff,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Aa)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Aa)),B) ) ) ).

% abs_less_iff
tff(fact_6897_abs__less__iff,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Aa)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),B) ) ) ).

% abs_less_iff
tff(fact_6898_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_6899_dense__eq0__I,axiom,
    ! [Xb: 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,aa_real_real(abs_abs_real,Xb)),E2) )
     => ( Xb = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_6900_dense__eq0__I,axiom,
    ! [Xb: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),E2) )
     => ( Xb = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_6901_abs__mult__pos,axiom,
    ! [Xb: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y3)),Xb) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6902_abs__mult__pos,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(times_times_real(aa_real_real(abs_abs_real,Y3)),Xb) = aa_real_real(abs_abs_real,aa_real_real(times_times_real(Y3),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6903_abs__mult__pos,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Y3)),Xb) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y3),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6904_abs__mult__pos,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y3)),Xb) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6905_abs__eq__mult,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Aa)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_6906_abs__eq__mult,axiom,
    ! [Aa: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(times_times_real(Aa),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_6907_abs__eq__mult,axiom,
    ! [Aa: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Aa),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_6908_abs__eq__mult,axiom,
    ! [Aa: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_6909_abs__div__pos,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
     => ( aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,Xb)),Y3) = aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Xb),Y3)) ) ) ).

% abs_div_pos
tff(fact_6910_abs__div__pos,axiom,
    ! [Y3: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y3)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Xb)),Y3) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Xb),Y3)) ) ) ).

% abs_div_pos
tff(fact_6911_eq__abs__iff_H,axiom,
    ! [Aa: real,B: real] :
      ( ( Aa = aa_real_real(abs_abs_real,B) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
        & ( ( B = Aa )
          | ( B = aa_real_real(uminus_uminus_real,Aa) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6912_eq__abs__iff_H,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( Aa = abs_abs_Code_integer(B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
        & ( ( B = Aa )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6913_eq__abs__iff_H,axiom,
    ! [Aa: rat,B: rat] :
      ( ( Aa = aa_rat_rat(abs_abs_rat,B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
        & ( ( B = Aa )
          | ( B = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6914_eq__abs__iff_H,axiom,
    ! [Aa: int,B: int] :
      ( ( Aa = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
        & ( ( B = Aa )
          | ( B = aa_int_int(uminus_uminus_int,Aa) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6915_abs__eq__iff_H,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_real(abs_abs_real,Aa) = B )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
        & ( ( Aa = B )
          | ( Aa = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6916_abs__eq__iff_H,axiom,
    ! [Aa: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(Aa) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( Aa = B )
          | ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6917_abs__eq__iff_H,axiom,
    ! [Aa: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,Aa) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( Aa = B )
          | ( Aa = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6918_abs__eq__iff_H,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(abs_abs_int,Aa) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( Aa = B )
          | ( Aa = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6919_abs__minus__le__zero,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,Aa))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_6920_abs__minus__le__zero,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(Aa))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_6921_abs__minus__le__zero,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,Aa))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_6922_abs__minus__le__zero,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,Aa))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_6923_zero__le__power__abs,axiom,
    ! [Aa: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(Aa)),Nb)) ).

% zero_le_power_abs
tff(fact_6924_zero__le__power__abs,axiom,
    ! [Aa: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Aa)),Nb)) ).

% zero_le_power_abs
tff(fact_6925_zero__le__power__abs,axiom,
    ! [Aa: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,Aa)),Nb)) ).

% zero_le_power_abs
tff(fact_6926_zero__le__power__abs,axiom,
    ! [Aa: 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,aa_int_int(abs_abs_int,Aa)),Nb)) ).

% zero_le_power_abs
tff(fact_6927_abs__if__raw,axiom,
    ! [X3: int] :
      aa_int_int(abs_abs_int,X3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X3),zero_zero_int),aa_int_int(uminus_uminus_int,X3),X3) ).

% abs_if_raw
tff(fact_6928_abs__if__raw,axiom,
    ! [X3: real] :
      aa_real_real(abs_abs_real,X3) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real),aa_real_real(uminus_uminus_real,X3),X3) ).

% abs_if_raw
tff(fact_6929_abs__if__raw,axiom,
    ! [X3: rat] :
      aa_rat_rat(abs_abs_rat,X3) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X3),X3) ).

% abs_if_raw
tff(fact_6930_abs__if__raw,axiom,
    ! [X3: code_integer] :
      abs_abs_Code_integer(X3) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3),X3) ).

% abs_if_raw
tff(fact_6931_abs__of__neg,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_int(abs_abs_int,Aa) = aa_int_int(uminus_uminus_int,Aa) ) ) ).

% abs_of_neg
tff(fact_6932_abs__of__neg,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_real(abs_abs_real,Aa) = aa_real_real(uminus_uminus_real,Aa) ) ) ).

% abs_of_neg
tff(fact_6933_abs__of__neg,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,Aa) = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ).

% abs_of_neg
tff(fact_6934_abs__of__neg,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(Aa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ) ) ).

% abs_of_neg
tff(fact_6935_abs__if,axiom,
    ! [Aa: int] :
      aa_int_int(abs_abs_int,Aa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int),aa_int_int(uminus_uminus_int,Aa),Aa) ).

% abs_if
tff(fact_6936_abs__if,axiom,
    ! [Aa: real] :
      aa_real_real(abs_abs_real,Aa) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real),aa_real_real(uminus_uminus_real,Aa),Aa) ).

% abs_if
tff(fact_6937_abs__if,axiom,
    ! [Aa: rat] :
      aa_rat_rat(abs_abs_rat,Aa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,Aa),Aa) ).

% abs_if
tff(fact_6938_abs__if,axiom,
    ! [Aa: code_integer] :
      abs_abs_Code_integer(Aa) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),Aa) ).

% abs_if
tff(fact_6939_abs__triangle__ineq4,axiom,
    ! [Aa: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Aa,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(Aa),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_6940_abs__triangle__ineq4,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Aa),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_6941_abs__triangle__ineq4,axiom,
    ! [Aa: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Aa),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_6942_abs__triangle__ineq4,axiom,
    ! [Aa: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_6943_abs__diff__triangle__ineq,axiom,
    ! [Aa: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(Aa,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(Aa,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_6944_abs__diff__triangle__ineq,axiom,
    ! [Aa: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Aa),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Aa),C))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6945_abs__diff__triangle__ineq,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),B)),aa_rat_rat(plus_plus_rat(C),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Aa),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6946_abs__diff__triangle__ineq,axiom,
    ! [Aa: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6947_abs__diff__le__iff,axiom,
    ! [Xb: code_integer,Aa: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,Aa))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(Aa,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),plus_p5714425477246183910nteger(Aa,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6948_abs__diff__le__iff,axiom,
    ! [Xb: real,Aa: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Aa))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Aa),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(plus_plus_real(Aa),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6949_abs__diff__le__iff,axiom,
    ! [Xb: rat,Aa: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),Aa))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Aa),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(Aa),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6950_abs__diff__le__iff,axiom,
    ! [Xb: int,Aa: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Aa))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6951_abs__diff__less__iff,axiom,
    ! [Xb: code_integer,Aa: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,Aa))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(Aa,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),plus_p5714425477246183910nteger(Aa,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6952_abs__diff__less__iff,axiom,
    ! [Xb: real,Aa: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Aa))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Aa),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(Aa),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6953_abs__diff__less__iff,axiom,
    ! [Xb: rat,Aa: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),Aa))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Aa),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(Aa),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6954_abs__diff__less__iff,axiom,
    ! [Xb: int,Aa: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Aa))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6955_abs__real__def,axiom,
    ! [Aa: real] :
      aa_real_real(abs_abs_real,Aa) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real),aa_real_real(uminus_uminus_real,Aa),Aa) ).

% abs_real_def
tff(fact_6956_sin__bound__lemma,axiom,
    ! [Xb: real,Y3: real,U: real,V: real] :
      ( ( Xb = Y3 )
     => ( 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(minus_minus_real(aa_real_real(plus_plus_real(Xb),U)),Y3))),V) ) ) ).

% sin_bound_lemma
tff(fact_6957_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_6958_abs__add__one__gt__zero,axiom,
    ! [Xb: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xb))) ).

% abs_add_one_gt_zero
tff(fact_6959_abs__add__one__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),aa_real_real(abs_abs_real,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6960_abs__add__one__gt__zero,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6961_abs__add__one__gt__zero,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6962_of__int__leD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_leD
tff(fact_6963_of__int__leD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% of_int_leD
tff(fact_6964_of__int__leD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% of_int_leD
tff(fact_6965_of__int__leD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xb) ) ) ).

% of_int_leD
tff(fact_6966_of__int__lessD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_lessD
tff(fact_6967_of__int__lessD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% of_int_lessD
tff(fact_6968_of__int__lessD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% of_int_lessD
tff(fact_6969_of__int__lessD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xb) ) ) ).

% of_int_lessD
tff(fact_6970_lemma__interval,axiom,
    ! [Aa: real,Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [Y2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Y2))),D3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_6971_abs__le__square__iff,axiom,
    ! [Xb: code_integer,Y3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),abs_abs_Code_integer(Y3))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6972_abs__le__square__iff,axiom,
    ! [Xb: real,Y3: 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,Y3))
    <=> 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6973_abs__le__square__iff,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),aa_rat_rat(abs_abs_rat,Y3))
    <=> 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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6974_abs__le__square__iff,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Y3))
    <=> 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6975_abs__square__eq__1,axiom,
    ! [Xb: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(Xb) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_6976_abs__square__eq__1,axiom,
    ! [Xb: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,Xb) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_6977_abs__square__eq__1,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(abs_abs_real,Xb) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_6978_abs__square__eq__1,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,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_6979_power__even__abs,axiom,
    ! [Nb: nat,Aa: 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_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,Aa)),Nb) = aa_nat_rat(power_power_rat(Aa),Nb) ) ) ).

% power_even_abs
tff(fact_6980_power__even__abs,axiom,
    ! [Nb: nat,Aa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(Aa)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb) ) ) ).

% power_even_abs
tff(fact_6981_power__even__abs,axiom,
    ! [Nb: nat,Aa: 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(abs_abs_real,Aa)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb) ) ) ).

% power_even_abs
tff(fact_6982_power__even__abs,axiom,
    ! [Nb: nat,Aa: 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_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,Aa)),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb) ) ) ).

% power_even_abs
tff(fact_6983_power2__le__iff__abs__le,axiom,
    ! [Y3: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),Y3) ) ) ).

% power2_le_iff_abs_le
tff(fact_6984_power2__le__iff__abs__le,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),Y3) ) ) ).

% power2_le_iff_abs_le
tff(fact_6985_power2__le__iff__abs__le,axiom,
    ! [Y3: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y3)
     => ( 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(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),Y3) ) ) ).

% power2_le_iff_abs_le
tff(fact_6986_power2__le__iff__abs__le,axiom,
    ! [Y3: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
     => ( 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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),Y3) ) ) ).

% power2_le_iff_abs_le
tff(fact_6987_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xb: code_integer] :
      ( ! [X2: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X2),aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6988_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xb: 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(P,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6989_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xb: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X2)
         => aa_rat_o(aa_rat_fun_rat_o(P,X2),aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,aa_rat_rat(abs_abs_rat,Xb)),aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6990_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xb: int] :
      ( ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X2)
         => aa_int_o(aa_int_fun_int_o(P,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6991_abs__square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_6992_abs__square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_le_1
tff(fact_6993_abs__square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_6994_abs__square__le__1,axiom,
    ! [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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_le_1
tff(fact_6995_abs__square__less__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_6996_abs__square__less__1,axiom,
    ! [Xb: 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)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_less_1
tff(fact_6997_abs__square__less__1,axiom,
    ! [Xb: 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)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_6998_abs__square__less__1,axiom,
    ! [Xb: 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)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_less_1
tff(fact_6999_power__mono__even,axiom,
    ! [Nb: nat,Aa: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Aa)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_even
tff(fact_7000_power__mono__even,axiom,
    ! [Nb: nat,Aa: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_even
tff(fact_7001_power__mono__even,axiom,
    ! [Nb: nat,Aa: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_even
tff(fact_7002_power__mono__even,axiom,
    ! [Nb: nat,Aa: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Aa),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono_even
tff(fact_7003_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_1: nat] : aa_nat_o(P,X_1)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_7004_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_7005_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_7006_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_7007_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_7008_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_7009_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_7010_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = $ite(
        ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_7011_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          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))),Mb)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_7012_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(minus_minus_real(aa_real_real(ln_ln_real,aa_real_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_7013_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),Xb))),Xb))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_7014_round__unique,axiom,
    ! [Xb: real,Y3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y3)),aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xb) = Y3 ) ) ) ).

% round_unique
tff(fact_7015_round__unique,axiom,
    ! [Xb: rat,Y3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y3))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,Xb) = Y3 ) ) ) ).

% round_unique
tff(fact_7016_pred__subset__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ord_le2909477399203599485un_o_o(aTP_Lamm_hy(R),aTP_Lamm_hy(S3))
    <=> ord_le8980329558974975238eger_o(R,S3) ) ).

% pred_subset_eq2
tff(fact_7017_pred__subset__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ord_le3306795289226699318_num_o(aTP_Lamm_hz(R),aTP_Lamm_hz(S3))
    <=> ord_le880128212290418581um_num(R,S3) ) ).

% pred_subset_eq2
tff(fact_7018_pred__subset__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ord_le1526520017675560704_num_o(aTP_Lamm_ia(R),aTP_Lamm_ia(S3))
    <=> ord_le8085105155179020875at_num(R,S3) ) ).

% pred_subset_eq2
tff(fact_7019_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_ib(R),aTP_Lamm_ib(S3))
    <=> ord_le3146513528884898305at_nat(R,S3) ) ).

% pred_subset_eq2
tff(fact_7020_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_ic(R),aTP_Lamm_ic(S3))
    <=> ord_le2843351958646193337nt_int(R,S3) ) ).

% pred_subset_eq2
tff(fact_7021_round__unique_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xb) = Nb ) ) ).

% round_unique'
tff(fact_7022_round__unique_H,axiom,
    ! [Xb: rat,Nb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),aa_int_rat(ring_1_of_int_rat,Nb)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,Xb) = Nb ) ) ).

% round_unique'
tff(fact_7023_of__int__round__abs__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),Xb))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7024_of__int__round__abs__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),Xb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7025_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y3 )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y3 = $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_7026_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_7027_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_7028_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_7029_round__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_7030_round__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_7031_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_7032_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_7033_round__neg__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_7034_round__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_7035_abs__zmult__eq__1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Mb) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7036_round__mono,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xb)),aa_rat_int(archim7778729529865785530nd_rat,Y3)) ) ).

% round_mono
tff(fact_7037_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_7038_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_7039_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)),Mb)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7040_bot__empty__eq2,axiom,
    ! [X3: code_integer,Xa: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(bot_bo897209107778963633un_o_o,X3),(Xa))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X3,(Xa)),bot_bo5379713665208646970eger_o) ) ).

% bot_empty_eq2
tff(fact_7041_bot__empty__eq2,axiom,
    ! [X3: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(bot_bo3657453288633029378_num_o,X3),Xa)
    <=> member7279096912039735102um_num(product_Pair_num_num(X3,Xa),bot_bo9056780473022590049um_num) ) ).

% bot_empty_eq2
tff(fact_7042_bot__empty__eq2,axiom,
    ! [X3: nat,Xa: num] :
      ( aa_num_o(aa_nat_fun_num_o(bot_bo1877178017081890764_num_o,X3),Xa)
    <=> member9148766508732265716at_num(product_Pair_nat_num(X3,Xa),bot_bo7038385379056416535at_num) ) ).

% bot_empty_eq2
tff(fact_7043_bot__empty__eq2,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X3),Xa)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Xa),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_7044_bot__empty__eq2,axiom,
    ! [X3: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X3),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_7045_pred__equals__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ! [X: code_integer,Xa3: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X,(Xa3)),R)
        <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X,(Xa3)),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7046_pred__equals__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ! [X: num,Xa3: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X,Xa3),R)
        <=> member7279096912039735102um_num(product_Pair_num_num(X,Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7047_pred__equals__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ! [X: nat,Xa3: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X,Xa3),R)
        <=> member9148766508732265716at_num(product_Pair_nat_num(X,Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7048_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Xa3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa3),R)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7049_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X: int,Xa3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa3),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7050_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_7051_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_7052_round__diff__minimal,axiom,
    ! [Z: real,Mb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),aa_int_real(ring_1_of_int_real,Mb)))) ).

% round_diff_minimal
tff(fact_7053_round__diff__minimal,axiom,
    ! [Z: rat,Mb: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,Mb)))) ).

% round_diff_minimal
tff(fact_7054_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7055_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_7056_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_7057_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7058_subrelI,axiom,
    ! [R2: set_Pr448751882837621926eger_o,S: set_Pr448751882837621926eger_o] :
      ( ! [X2: code_integer,Y: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X2,(Y)),R2)
         => member1379723562493234055eger_o(produc6677183202524767010eger_o(X2,(Y)),S) )
     => ord_le8980329558974975238eger_o(R2,S) ) ).

% subrelI
tff(fact_7059_subrelI,axiom,
    ! [R2: set_Pr8218934625190621173um_num,S: set_Pr8218934625190621173um_num] :
      ( ! [X2: num,Y: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X2,Y),R2)
         => member7279096912039735102um_num(product_Pair_num_num(X2,Y),S) )
     => ord_le880128212290418581um_num(R2,S) ) ).

% subrelI
tff(fact_7060_subrelI,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S: set_Pr6200539531224447659at_num] :
      ( ! [X2: nat,Y: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X2,Y),R2)
         => member9148766508732265716at_num(product_Pair_nat_num(X2,Y),S) )
     => ord_le8085105155179020875at_num(R2,S) ) ).

% subrelI
tff(fact_7061_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y),S) )
     => ord_le3146513528884898305at_nat(R2,S) ) ).

% subrelI
tff(fact_7062_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X2: int,Y: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y),S) )
     => ord_le2843351958646193337nt_int(R2,S) ) ).

% subrelI
tff(fact_7063_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7064_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,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K3),L3))
           => ( ( ~ ( 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(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(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,L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K3),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_7065_pred__subset__eq,axiom,
    ! [R: set_complex,S3: set_complex] :
      ( ord_le7644322562001619171plex_o(aTP_Lamm_ab(R),aTP_Lamm_ab(S3))
    <=> aa_set_complex_o(ord_le211207098394363844omplex(R),S3) ) ).

% pred_subset_eq
tff(fact_7066_pred__subset__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_a(R),aTP_Lamm_a(S3))
    <=> aa_set_real_o(ord_less_eq_set_real(R),S3) ) ).

% pred_subset_eq
tff(fact_7067_pred__subset__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ac(R),aTP_Lamm_ac(S3))
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(R),S3) ) ).

% pred_subset_eq
tff(fact_7068_pred__subset__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ad(R),aTP_Lamm_ad(S3))
    <=> aa_set_nat_o(ord_less_eq_set_nat(R),S3) ) ).

% pred_subset_eq
tff(fact_7069_pred__subset__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ae(R),aTP_Lamm_ae(S3))
    <=> aa_set_int_o(ord_less_eq_set_int(R),S3) ) ).

% pred_subset_eq
tff(fact_7070_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(plus_plus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_7071_of__int__round__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_7072_of__int__round__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_ge
tff(fact_7073_of__int__round__ge,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))) ).

% of_int_round_ge
tff(fact_7074_of__int__round__gt,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_gt
tff(fact_7075_of__int__round__gt,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))) ).

% of_int_round_gt
tff(fact_7076_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [I2: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_7077_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_7078_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_7079_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_7080_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_7081_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_7082_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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,Xb)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),aa_real_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_7083_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_7084_add__scale__eq__noteq,axiom,
    ! [R2: complex,Aa: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( Aa = B )
          & ( C != D ) )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),C)) != aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_7085_add__scale__eq__noteq,axiom,
    ! [R2: real,Aa: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( Aa = B )
          & ( C != D ) )
       => ( aa_real_real(plus_plus_real(Aa),aa_real_real(times_times_real(R2),C)) != aa_real_real(plus_plus_real(B),aa_real_real(times_times_real(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_7086_add__scale__eq__noteq,axiom,
    ! [R2: rat,Aa: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( Aa = B )
          & ( C != D ) )
       => ( aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(times_times_rat(R2),C)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_7087_add__scale__eq__noteq,axiom,
    ! [R2: nat,Aa: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( Aa = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_7088_add__scale__eq__noteq,axiom,
    ! [R2: int,Aa: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( Aa = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_7089_Sum__Icc__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_aj,set_or1266510415728281911st_int(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_7090_or_Oidem,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),Aa) = Aa ).

% or.idem
tff(fact_7091_or_Oidem,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),Aa) = Aa ).

% or.idem
tff(fact_7092_or_Oleft__idem,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),B)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),B) ).

% or.left_idem
tff(fact_7093_or_Oleft__idem,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),B) ).

% or.left_idem
tff(fact_7094_or_Oright__idem,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),B)),B) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),B) ).

% or.right_idem
tff(fact_7095_or_Oright__idem,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),B) ).

% or.right_idem
tff(fact_7096_or_Oleft__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,zero_zero_int),Aa) = Aa ).

% or.left_neutral
tff(fact_7097_or_Oleft__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,zero_zero_nat),Aa) = Aa ).

% or.left_neutral
tff(fact_7098_or_Oright__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),zero_zero_int) = Aa ).

% or.right_neutral
tff(fact_7099_or_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),zero_zero_nat) = Aa ).

% or.right_neutral
tff(fact_7100_sum_Oneutral__const,axiom,
    ! [A: set_int] : groups4538972089207619220nt_int(aTP_Lamm_cv,A) = zero_zero_int ).

% sum.neutral_const
tff(fact_7101_sum_Oneutral__const,axiom,
    ! [A: set_complex] : groups7754918857620584856omplex(aTP_Lamm_cr,A) = zero_zero_complex ).

% sum.neutral_const
tff(fact_7102_sum_Oneutral__const,axiom,
    ! [A: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cu),A) = zero_zero_nat ).

% sum.neutral_const
tff(fact_7103_sum_Oneutral__const,axiom,
    ! [A: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_id),A) = zero_zero_real ).

% sum.neutral_const
tff(fact_7104_abs__sum__abs,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_int(abs_abs_int,groups4538972089207619220nt_int(aTP_Lamm_ie(F),A)) = groups4538972089207619220nt_int(aTP_Lamm_ie(F),A) ).

% abs_sum_abs
tff(fact_7105_abs__sum__abs,axiom,
    ! [F: fun_nat_real,A: set_nat] : aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_if(F)),A)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_if(F)),A) ).

% abs_sum_abs
tff(fact_7106_bit_Odisj__one__left,axiom,
    ! [Xb: code_integer] : bit_se1080825931792720795nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Xb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_left
tff(fact_7107_bit_Odisj__one__left,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_left
tff(fact_7108_bit_Odisj__one__right,axiom,
    ! [Xb: code_integer] : bit_se1080825931792720795nteger(Xb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_right
tff(fact_7109_bit_Odisj__one__right,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_right
tff(fact_7110_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_7111_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_7112_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_7113_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_7114_sum_Odelta_H,axiom,
    ! [S3: set_real,Aa: real,B: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ig(Aa),B),S3) = $ite(aa_set_real_o(member_real(Aa),S3),aa_real_complex(B,Aa),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_7115_sum_Odelta_H,axiom,
    ! [S3: set_nat,Aa: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ih(Aa),B)),S3) = $ite(aa_set_nat_o(member_nat(Aa),S3),aa_nat_complex(B,Aa),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_7116_sum_Odelta_H,axiom,
    ! [S3: set_int,Aa: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ii(Aa),B),S3) = $ite(aa_set_int_o(member_int(Aa),S3),aa_int_complex(B,Aa),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_7117_sum_Odelta_H,axiom,
    ! [S3: set_real,Aa: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ij(Aa),B),S3) = $ite(aa_set_real_o(member_real(Aa),S3),aa_real_real(B,Aa),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7118_sum_Odelta_H,axiom,
    ! [S3: set_int,Aa: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ik(Aa),B),S3) = $ite(aa_set_int_o(member_int(Aa),S3),aa_int_real(B,Aa),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7119_sum_Odelta_H,axiom,
    ! [S3: set_complex,Aa: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_il(Aa),B),S3) = $ite(aa_set_complex_o(member_complex(Aa),S3),aa_complex_real(B,Aa),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7120_sum_Odelta_H,axiom,
    ! [S3: set_real,Aa: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_im(Aa),B),S3) = $ite(aa_set_real_o(member_real(Aa),S3),aa_real_rat(B,Aa),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7121_sum_Odelta_H,axiom,
    ! [S3: set_nat,Aa: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_in(Aa),B),S3) = $ite(aa_set_nat_o(member_nat(Aa),S3),aa_nat_rat(B,Aa),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7122_sum_Odelta_H,axiom,
    ! [S3: set_int,Aa: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_io(Aa),B),S3) = $ite(aa_set_int_o(member_int(Aa),S3),aa_int_rat(B,Aa),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7123_sum_Odelta_H,axiom,
    ! [S3: set_complex,Aa: complex,B: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ip(Aa),B),S3) = $ite(aa_set_complex_o(member_complex(Aa),S3),aa_complex_rat(B,Aa),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7124_sum_Odelta,axiom,
    ! [S3: set_real,Aa: real,B: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_iq(Aa),B),S3) = $ite(aa_set_real_o(member_real(Aa),S3),aa_real_complex(B,Aa),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_7125_sum_Odelta,axiom,
    ! [S3: set_nat,Aa: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ir(Aa),B)),S3) = $ite(aa_set_nat_o(member_nat(Aa),S3),aa_nat_complex(B,Aa),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_7126_sum_Odelta,axiom,
    ! [S3: set_int,Aa: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_is(Aa),B),S3) = $ite(aa_set_int_o(member_int(Aa),S3),aa_int_complex(B,Aa),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_7127_sum_Odelta,axiom,
    ! [S3: set_real,Aa: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_it(Aa),B),S3) = $ite(aa_set_real_o(member_real(Aa),S3),aa_real_real(B,Aa),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7128_sum_Odelta,axiom,
    ! [S3: set_int,Aa: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_iu(Aa),B),S3) = $ite(aa_set_int_o(member_int(Aa),S3),aa_int_real(B,Aa),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7129_sum_Odelta,axiom,
    ! [S3: set_complex,Aa: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_iv(Aa),B),S3) = $ite(aa_set_complex_o(member_complex(Aa),S3),aa_complex_real(B,Aa),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7130_sum_Odelta,axiom,
    ! [S3: set_real,Aa: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_iw(Aa),B),S3) = $ite(aa_set_real_o(member_real(Aa),S3),aa_real_rat(B,Aa),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7131_sum_Odelta,axiom,
    ! [S3: set_nat,Aa: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ix(Aa),B),S3) = $ite(aa_set_nat_o(member_nat(Aa),S3),aa_nat_rat(B,Aa),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7132_sum_Odelta,axiom,
    ! [S3: set_int,Aa: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_iy(Aa),B),S3) = $ite(aa_set_int_o(member_int(Aa),S3),aa_int_rat(B,Aa),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7133_sum_Odelta,axiom,
    ! [S3: set_complex,Aa: complex,B: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_iz(Aa),B),S3) = $ite(aa_set_complex_o(member_complex(Aa),S3),aa_complex_rat(B,Aa),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7134_sum__abs,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A))),groups4538972089207619220nt_int(aTP_Lamm_ie(F),A)) ).

% sum_abs
tff(fact_7135_sum__abs,axiom,
    ! [F: fun_nat_real,A: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_if(F)),A)) ).

% sum_abs
tff(fact_7136_sum_Oinsert,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A)
       => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(Xb,A)) = aa_real_real(plus_plus_real(aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,A)) ) ) ) ).

% sum.insert
tff(fact_7137_sum_Oinsert,axiom,
    ! [A: set_real,Xb: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ~ aa_set_real_o(member_real(Xb),A)
       => ( groups8097168146408367636l_real(G,insert_real(Xb,A)) = aa_real_real(plus_plus_real(aa_real_real(G,Xb)),groups8097168146408367636l_real(G,A)) ) ) ) ).

% sum.insert
tff(fact_7138_sum_Oinsert,axiom,
    ! [A: set_int,Xb: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(Xb),A)
       => ( groups8778361861064173332t_real(G,insert_int(Xb,A)) = aa_real_real(plus_plus_real(aa_int_real(G,Xb)),groups8778361861064173332t_real(G,A)) ) ) ) ).

% sum.insert
tff(fact_7139_sum_Oinsert,axiom,
    ! [A: set_complex,Xb: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ~ aa_set_complex_o(member_complex(Xb),A)
       => ( groups5808333547571424918x_real(G,insert_complex(Xb,A)) = aa_real_real(plus_plus_real(aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,A)) ) ) ) ).

% sum.insert
tff(fact_7140_sum_Oinsert,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A)
       => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(Xb,A)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7141_sum_Oinsert,axiom,
    ! [A: set_real,Xb: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ~ aa_set_real_o(member_real(Xb),A)
       => ( groups1300246762558778688al_rat(G,insert_real(Xb,A)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xb)),groups1300246762558778688al_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7142_sum_Oinsert,axiom,
    ! [A: set_nat,Xb: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(Xb),A)
       => ( groups2906978787729119204at_rat(G,insert_nat(Xb,A)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xb)),groups2906978787729119204at_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7143_sum_Oinsert,axiom,
    ! [A: set_int,Xb: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(Xb),A)
       => ( groups3906332499630173760nt_rat(G,insert_int(Xb,A)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7144_sum_Oinsert,axiom,
    ! [A: set_complex,Xb: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ~ aa_set_complex_o(member_complex(Xb),A)
       => ( groups5058264527183730370ex_rat(G,insert_complex(Xb,A)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7145_sum_Oinsert,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A)
       => ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(Xb,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7146_or__numerals_I8_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb)) ).

% or_numerals(8)
tff(fact_7147_or__numerals_I8_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))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_numerals(8)
tff(fact_7148_or__numerals_I2_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3)) ).

% or_numerals(2)
tff(fact_7149_or__numerals_I2_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_numerals(2)
tff(fact_7150_of__int__sum,axiom,
    ! [F: fun_complex_int,A: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A)) = groups7754918857620584856omplex(aTP_Lamm_ja(F),A) ).

% of_int_sum
tff(fact_7151_of__int__sum,axiom,
    ! [F: fun_nat_int,A: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jb(F)),A) ).

% of_int_sum
tff(fact_7152_of__int__sum,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A)) = groups8778361861064173332t_real(aTP_Lamm_jc(F),A) ).

% of_int_sum
tff(fact_7153_of__int__sum,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A)) = groups3906332499630173760nt_rat(aTP_Lamm_jd(F),A) ).

% of_int_sum
tff(fact_7154_of__int__sum,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A)) = groups4538972089207619220nt_int(aTP_Lamm_je(F),A) ).

% of_int_sum
tff(fact_7155_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_ie(F),A)) ).

% sum_abs_ge_zero
tff(fact_7156_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_if(F)),A)) ).

% sum_abs_ge_zero
tff(fact_7157_or__numerals_I3_J,axiom,
    ! [Xb: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y3))) ).

% or_numerals(3)
tff(fact_7158_or__numerals_I3_J,axiom,
    ! [Xb: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y3))) ).

% or_numerals(3)
tff(fact_7159_or__numerals_I5_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb)) ).

% or_numerals(5)
tff(fact_7160_or__numerals_I5_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))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_numerals(5)
tff(fact_7161_or__numerals_I1_J,axiom,
    ! [Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3)) ).

% or_numerals(1)
tff(fact_7162_or__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_numerals(1)
tff(fact_7163_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_7164_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_7165_or__numerals_I7_J,axiom,
    ! [Xb: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y3)))) ).

% or_numerals(7)
tff(fact_7166_or__numerals_I7_J,axiom,
    ! [Xb: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% or_numerals(7)
tff(fact_7167_or__numerals_I6_J,axiom,
    ! [Xb: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y3)))) ).

% or_numerals(6)
tff(fact_7168_or__numerals_I6_J,axiom,
    ! [Xb: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% or_numerals(6)
tff(fact_7169_or__numerals_I4_J,axiom,
    ! [Xb: num,Y3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y3)))) ).

% or_numerals(4)
tff(fact_7170_or__numerals_I4_J,axiom,
    ! [Xb: num,Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y3)))) ).

% or_numerals(4)
tff(fact_7171_bit_Odisj__zero__right,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),zero_zero_int) = Xb ).

% bit.disj_zero_right
tff(fact_7172_or__eq__0__iff,axiom,
    ! [Aa: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),B) = zero_zero_int )
    <=> ( ( Aa = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
tff(fact_7173_or__eq__0__iff,axiom,
    ! [Aa: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),B) = zero_zero_nat )
    <=> ( ( Aa = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
tff(fact_7174_or_Oassoc,axiom,
    ! [Aa: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),B)),C) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,B),C)) ).

% or.assoc
tff(fact_7175_or_Oassoc,axiom,
    ! [Aa: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),C)) ).

% or.assoc
tff(fact_7176_or_Ocommute,axiom,
    ! [Aa: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),B) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,B),Aa) ).

% or.commute
tff(fact_7177_or_Ocommute,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),Aa) ).

% or.commute
tff(fact_7178_or_Oleft__commute,axiom,
    ! [B: int,Aa: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,B),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),C)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,B),C)) ).

% or.left_commute
tff(fact_7179_or_Oleft__commute,axiom,
    ! [B: nat,Aa: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),C)) ).

% or.left_commute
tff(fact_7180_sum_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B2: set_int,A: set_int] : groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_jf(G),B2),A) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_jh(G),A),B2) ).

% sum.swap
tff(fact_7181_sum_Oswap,axiom,
    ! [G: fun_co7673397065115134084omplex,B2: set_complex,A: set_complex] : groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_ji(G),B2),A) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_jk(G),A),B2) ).

% sum.swap
tff(fact_7182_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B2: set_nat,A: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_jl(G),B2)),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_jn(G),A)),B2) ).

% sum.swap
tff(fact_7183_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_real,B2: set_nat,A: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_jo(G),B2)),A) = aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_jq(G),A)),B2) ).

% sum.swap
tff(fact_7184_of__int__or__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_or_eq
tff(fact_7185_bit_Oconj__disj__distrib,axiom,
    ! [Xb: int,Y3: int,Z: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Y3),Z)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y3)),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Z)) ).

% bit.conj_disj_distrib
tff(fact_7186_bit_Odisj__conj__distrib,axiom,
    ! [Xb: int,Y3: int,Z: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y3),Z)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y3)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Z)) ).

% bit.disj_conj_distrib
tff(fact_7187_bit_Oconj__disj__distrib2,axiom,
    ! [Y3: int,Z: int,Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Y3),Z)),Xb) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y3),Xb)),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Z),Xb)) ).

% bit.conj_disj_distrib2
tff(fact_7188_bit_Odisj__conj__distrib2,axiom,
    ! [Y3: int,Z: int,Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y3),Z)),Xb) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Y3),Xb)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Z),Xb)) ).

% bit.disj_conj_distrib2
tff(fact_7189_arctan__monotone_H,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y3)) ) ).

% arctan_monotone'
tff(fact_7190_arctan__le__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ).

% arctan_le_iff
tff(fact_7191_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5)) ) ).

% sum_mono
tff(fact_7192_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5)) ) ).

% sum_mono
tff(fact_7193_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5)) ) ).

% sum_mono
tff(fact_7194_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_7195_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5)) ) ).

% sum_mono
tff(fact_7196_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5)) ) ).

% sum_mono
tff(fact_7197_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_7198_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,K5)),groups5690904116761175830ex_int(G,K5)) ) ).

% sum_mono
tff(fact_7199_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5)) ) ).

% sum_mono
tff(fact_7200_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5)) ) ).

% sum_mono
tff(fact_7201_sum__product,axiom,
    ! [F: fun_int_int,A: set_int,G: fun_int_int,B2: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A)),groups4538972089207619220nt_int(G,B2)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_js(F),G),B2),A) ).

% sum_product
tff(fact_7202_sum__product,axiom,
    ! [F: fun_complex_complex,A: set_complex,G: fun_complex_complex,B2: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A)),groups7754918857620584856omplex(G,B2)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_ju(F),G),B2),A) ).

% sum_product
tff(fact_7203_sum__product,axiom,
    ! [F: fun_nat_nat,A: set_nat,G: fun_nat_nat,B2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_jw(F),G),B2)),A) ).

% sum_product
tff(fact_7204_sum__product,axiom,
    ! [F: fun_nat_real,A: set_nat,G: fun_nat_real,B2: set_nat] : aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(F),A)),aa_set_nat_real(groups6591440286371151544t_real(G),B2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_jy(F),G),B2)),A) ).

% sum_product
tff(fact_7205_sum__distrib__right,axiom,
    ! [F: fun_int_int,A: set_int,R2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_jz(F),R2),A) ).

% sum_distrib_right
tff(fact_7206_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A: set_complex,R2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_ka(F),R2),A) ).

% sum_distrib_right
tff(fact_7207_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kb(F),R2)),A) ).

% sum_distrib_right
tff(fact_7208_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A: set_nat,R2: real] : aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(F),A)),R2) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),R2)),A) ).

% sum_distrib_right
tff(fact_7209_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R2),groups4538972089207619220nt_int(F,A)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kd(R2),F),A) ).

% sum_distrib_left
tff(fact_7210_sum__distrib__left,axiom,
    ! [R2: complex,F: fun_complex_complex,A: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),groups7754918857620584856omplex(F,A)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ke(R2),F),A) ).

% sum_distrib_left
tff(fact_7211_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kf(R2),F)),A) ).

% sum_distrib_left
tff(fact_7212_sum__distrib__left,axiom,
    ! [R2: real,F: fun_nat_real,A: set_nat] : aa_real_real(times_times_real(R2),aa_set_nat_real(groups6591440286371151544t_real(F),A)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_kg(R2),F)),A) ).

% sum_distrib_left
tff(fact_7213_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kh(G),H),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A)),groups4538972089207619220nt_int(H,A)) ).

% sum.distrib
tff(fact_7214_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ki(G),H),A) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups7754918857620584856omplex(G,A)),groups7754918857620584856omplex(H,A)) ).

% sum.distrib
tff(fact_7215_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kj(G),H)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A)) ).

% sum.distrib
tff(fact_7216_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_kk(G),H)),A) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),A)),aa_set_nat_real(groups6591440286371151544t_real(H),A)) ).

% sum.distrib
tff(fact_7217_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kl(F),G),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A)),groups4538972089207619220nt_int(G,A)) ).

% sum_subtractf
tff(fact_7218_sum__subtractf,axiom,
    ! [F: fun_complex_complex,G: fun_complex_complex,A: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_km(F),G),A) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups7754918857620584856omplex(F,A)),groups7754918857620584856omplex(G,A)) ).

% sum_subtractf
tff(fact_7219_sum__subtractf,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,A: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_kn(F),G)),A) = aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(F),A)),aa_set_nat_real(groups6591440286371151544t_real(G),A)) ).

% sum_subtractf
tff(fact_7220_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A: set_complex,R2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups7754918857620584856omplex(F,A)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_ko(F),R2),A) ).

% sum_divide_distrib
tff(fact_7221_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A: set_nat,R2: real] : aa_real_real(divide_divide_real(aa_set_nat_real(groups6591440286371151544t_real(F),A)),R2) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kp(F),R2)),A) ).

% sum_divide_distrib
tff(fact_7222_sum__negf,axiom,
    ! [F: fun_int_int,A: set_int] : groups4538972089207619220nt_int(aTP_Lamm_kq(F),A) = aa_int_int(uminus_uminus_int,groups4538972089207619220nt_int(F,A)) ).

% sum_negf
tff(fact_7223_sum__negf,axiom,
    ! [F: fun_complex_complex,A: set_complex] : groups7754918857620584856omplex(aTP_Lamm_kr(F),A) = aa_complex_complex(uminus1482373934393186551omplex,groups7754918857620584856omplex(F,A)) ).

% sum_negf
tff(fact_7224_sum__negf,axiom,
    ! [F: fun_nat_real,A: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ks(F)),A) = aa_real_real(uminus_uminus_real,aa_set_nat_real(groups6591440286371151544t_real(F),A)) ).

% sum_negf
tff(fact_7225_sum_Oswap__restrict,axiom,
    ! [A: set_real,B2: set_int,G: fun_real_fun_int_int,R: fun_real_fun_int_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups1932886352136224148al_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_ku(B2),G),R),A) = groups4538972089207619220nt_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_kw(A),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7226_sum_Oswap__restrict,axiom,
    ! [A: set_nat,B2: set_int,G: fun_nat_fun_int_int,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups3539618377306564664at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_ky(B2),G),R),A) = groups4538972089207619220nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_la(A),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7227_sum_Oswap__restrict,axiom,
    ! [A: set_complex,B2: set_int,G: fun_co333780937270390912nt_int,R: fun_co3431063785546946928_int_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups5690904116761175830ex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_lc(B2),G),R),A) = groups4538972089207619220nt_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_lf(A),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7228_sum_Oswap__restrict,axiom,
    ! [A: set_real,B2: set_complex,G: fun_re7208826350674946562omplex,R: fun_re7319081385389557872plex_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5754745047067104278omplex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_lh(B2),G),R),A) = groups7754918857620584856omplex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_lj(A),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7229_sum_Oswap__restrict,axiom,
    ! [A: set_nat,B2: set_complex,G: fun_na1591161823130732966omplex,R: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_ll(B2),G),R)),A) = groups7754918857620584856omplex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_ln(A),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7230_sum_Oswap__restrict,axiom,
    ! [A: set_int,B2: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5065666251602127472plex_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups3049146728041665814omplex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_lp(B2),G),R),A) = groups7754918857620584856omplex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_lr(A),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7231_sum_Oswap__restrict,axiom,
    ! [A: set_real,B2: set_nat,G: fun_real_fun_nat_nat,R: fun_real_fun_nat_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups1935376822645274424al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_lt(B2),G),R),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_lv(A),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7232_sum_Oswap__restrict,axiom,
    ! [A: set_int,B2: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_lx(B2),G),R),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_lz(A),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7233_sum_Oswap__restrict,axiom,
    ! [A: set_complex,B2: set_nat,G: fun_co636942507509095880at_nat,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups5693394587270226106ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_mb(B2),G),R),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_md(A),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7234_sum_Oswap__restrict,axiom,
    ! [A: set_real,B2: set_nat,G: fun_re4238438383274954274t_real,R: fun_real_fun_nat_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups8097168146408367636l_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_me(B2),G),R),A) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_mg(A),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7235_mod__sum__eq,axiom,
    ! [F: fun_int_int,Aa: int,A: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_mh(F),Aa),A)),Aa) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A)),Aa) ).

% mod_sum_eq
tff(fact_7236_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,Aa: nat,A: 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_mi(F),Aa)),A)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),Aa) ).

% mod_sum_eq
tff(fact_7237_sum__nonneg,axiom,
    ! [A: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A)
         => 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_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A)) ) ).

% sum_nonneg
tff(fact_7238_sum__nonneg,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A)
         => 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,A)) ) ).

% sum_nonneg
tff(fact_7239_sum__nonneg,axiom,
    ! [A: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => 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,A)) ) ).

% sum_nonneg
tff(fact_7240_sum__nonneg,axiom,
    ! [A: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A)
         => 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_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A)) ) ).

% sum_nonneg
tff(fact_7241_sum__nonneg,axiom,
    ! [A: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A)
         => 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,A)) ) ).

% sum_nonneg
tff(fact_7242_sum__nonneg,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => 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,A)) ) ).

% sum_nonneg
tff(fact_7243_sum__nonneg,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => 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,A)) ) ).

% sum_nonneg
tff(fact_7244_sum__nonneg,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A)) ) ).

% sum_nonneg
tff(fact_7245_sum__nonneg,axiom,
    ! [A: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A)
         => 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,A)) ) ).

% sum_nonneg
tff(fact_7246_sum__nonneg,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => 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,A)) ) ).

% sum_nonneg
tff(fact_7247_sum__nonpos,axiom,
    ! [A: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7248_sum__nonpos,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A)
         => 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,A)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7249_sum__nonpos,axiom,
    ! [A: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => 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,A)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7250_sum__nonpos,axiom,
    ! [A: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7251_sum__nonpos,axiom,
    ! [A: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A)
         => 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,A)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7252_sum__nonpos,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => 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,A)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7253_sum__nonpos,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => 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,A)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7254_sum__nonpos,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7255_sum__nonpos,axiom,
    ! [A: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A)
         => 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,A)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7256_sum__nonpos,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => 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,A)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7257_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I6: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I6) = groups1300246762558778688al_rat(G,I6) )
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) )
       => ( aa_set_real_o(member_real(I),I6)
         => ( aa_set_real_o(finite_finite_real,I6)
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7258_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I6: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I6) = groups2906978787729119204at_rat(G,I6) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
       => ( aa_set_nat_o(member_nat(I),I6)
         => ( aa_set_nat_o(finite_finite_nat,I6)
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7259_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I6: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I6) = groups3906332499630173760nt_rat(G,I6) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I6)
         => ( aa_set_int_o(finite_finite_int,I6)
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7260_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I6: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I6) = groups5058264527183730370ex_rat(G,I6) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I6)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I6)
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7261_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I6: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I6) = groups1935376822645274424al_nat(G,I6) )
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
       => ( aa_set_real_o(member_real(I),I6)
         => ( aa_set_real_o(finite_finite_real,I6)
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7262_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I6: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I6) = groups4541462559716669496nt_nat(G,I6) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I6)
         => ( aa_set_int_o(finite_finite_int,I6)
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7263_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I6: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I6) = groups5693394587270226106ex_nat(G,I6) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I6)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I6)
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7264_sum__mono__inv,axiom,
    ! [F: fun_real_int,I6: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I6) = groups1932886352136224148al_int(G,I6) )
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) )
       => ( aa_set_real_o(member_real(I),I6)
         => ( aa_set_real_o(finite_finite_real,I6)
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7265_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I6: set_nat,G: fun_nat_int,I: nat] :
      ( ( groups3539618377306564664at_int(F,I6) = groups3539618377306564664at_int(G,I6) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
       => ( aa_set_nat_o(member_nat(I),I6)
         => ( aa_set_nat_o(finite_finite_nat,I6)
           => ( aa_nat_int(F,I) = aa_nat_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7266_sum__mono__inv,axiom,
    ! [F: fun_complex_int,I6: set_complex,G: fun_complex_int,I: complex] :
      ( ( groups5690904116761175830ex_int(F,I6) = groups5690904116761175830ex_int(G,I6) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I6)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I6)
           => ( aa_complex_int(F,I) = aa_complex_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7267_OR__lower,axiom,
    ! [Xb: int,Y3: 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),Y3)
       => 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),Y3)) ) ) ).

% OR_lower
tff(fact_7268_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_7269_plus__and__or,axiom,
    ! [Xb: int,Y3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y3)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y3) ).

% plus_and_or
tff(fact_7270_sum_Ointer__filter,axiom,
    ! [A: set_real,G: fun_real_complex,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bc(A),P))) = groups5754745047067104278omplex(aa_fun1098856492951973088omplex(aTP_Lamm_mj(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7271_sum_Ointer__filter,axiom,
    ! [A: set_nat,G: fun_nat_complex,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bg(A),P))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_fun1400689748890728928omplex(aTP_Lamm_mk(G),P)),A) ) ) ).

% sum.inter_filter
tff(fact_7272_sum_Ointer__filter,axiom,
    ! [A: set_int,G: fun_int_complex,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(A),P))) = groups3049146728041665814omplex(aa_fun6439487526662521568omplex(aTP_Lamm_ml(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7273_sum_Ointer__filter,axiom,
    ! [A: set_real,G: fun_real_real,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bc(A),P))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_mm(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7274_sum_Ointer__filter,axiom,
    ! [A: set_int,G: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(A),P))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_mn(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7275_sum_Ointer__filter,axiom,
    ! [A: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A),P))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_mo(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7276_sum_Ointer__filter,axiom,
    ! [A: set_real,G: fun_real_rat,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bc(A),P))) = groups1300246762558778688al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_mp(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7277_sum_Ointer__filter,axiom,
    ! [A: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bg(A),P))) = groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_mq(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7278_sum_Ointer__filter,axiom,
    ! [A: set_int,G: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(A),P))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_mr(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7279_sum_Ointer__filter,axiom,
    ! [A: set_complex,G: fun_complex_rat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A),P))) = groups5058264527183730370ex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_ms(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7280_sum__le__included,axiom,
    ! [S: set_int,Ta: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( 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),S)
               => ? [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,S)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7281_sum__le__included,axiom,
    ! [S: set_int,Ta: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( 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),S)
               => ? [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,S)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7282_sum__le__included,axiom,
    ! [S: set_complex,Ta: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( 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),S)
               => ? [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,S)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7283_sum__le__included,axiom,
    ! [S: set_complex,Ta: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( 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),S)
               => ? [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,S)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7284_sum__le__included,axiom,
    ! [S: set_nat,Ta: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( 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),S)
               => ? [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,S)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7285_sum__le__included,axiom,
    ! [S: set_nat,Ta: set_int,G: fun_int_rat,I: fun_int_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X2)) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_nat(I,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7286_sum__le__included,axiom,
    ! [S: set_nat,Ta: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X2)) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_nat(I,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7287_sum__le__included,axiom,
    ! [S: set_int,Ta: set_nat,G: fun_nat_rat,I: fun_nat_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( 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: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_int(I,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7288_sum__le__included,axiom,
    ! [S: set_int,Ta: set_int,G: fun_int_rat,I: fun_int_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X2)) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_int(I,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7289_sum__le__included,axiom,
    ! [S: set_int,Ta: set_complex,G: fun_complex_rat,I: fun_complex_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X2)) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_int(I,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7290_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
       => ( ( groups8097168146408367636l_real(F,A) = zero_zero_real )
        <=> ! [X: real] :
              ( aa_set_real_o(member_real(X),A)
             => ( aa_real_real(F,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7291_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2)) )
       => ( ( groups8778361861064173332t_real(F,A) = zero_zero_real )
        <=> ! [X: int] :
              ( aa_set_int_o(member_int(X),A)
             => ( aa_int_real(F,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7292_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2)) )
       => ( ( groups5808333547571424918x_real(F,A) = zero_zero_real )
        <=> ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A)
             => ( aa_complex_real(F,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7293_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)) )
       => ( ( groups1300246762558778688al_rat(F,A) = zero_zero_rat )
        <=> ! [X: real] :
              ( aa_set_real_o(member_real(X),A)
             => ( aa_real_rat(F,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7294_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
       => ( ( groups2906978787729119204at_rat(F,A) = zero_zero_rat )
        <=> ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A)
             => ( aa_nat_rat(F,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7295_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2)) )
       => ( ( groups3906332499630173760nt_rat(F,A) = zero_zero_rat )
        <=> ! [X: int] :
              ( aa_set_int_o(member_int(X),A)
             => ( aa_int_rat(F,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7296_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2)) )
       => ( ( groups5058264527183730370ex_rat(F,A) = zero_zero_rat )
        <=> ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A)
             => ( aa_complex_rat(F,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7297_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2)) )
       => ( ( groups1935376822645274424al_nat(F,A) = zero_zero_nat )
        <=> ! [X: real] :
              ( aa_set_real_o(member_real(X),A)
             => ( aa_real_nat(F,X) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7298_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X2)) )
       => ( ( groups4541462559716669496nt_nat(F,A) = zero_zero_nat )
        <=> ! [X: int] :
              ( aa_set_int_o(member_int(X),A)
             => ( aa_int_nat(F,X) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7299_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X2)) )
       => ( ( groups5693394587270226106ex_nat(F,A) = zero_zero_nat )
        <=> ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A)
             => ( aa_complex_nat(F,X) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7300_sum__strict__mono__ex1,axiom,
    ! [A: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
           => 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),A)
              & 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,A)),groups8778361861064173332t_real(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7301_sum__strict__mono__ex1,axiom,
    ! [A: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A)
           => 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),A)
              & 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,A)),groups5808333547571424918x_real(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7302_sum__strict__mono__ex1,axiom,
    ! [A: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A)
           => 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),A)
              & 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,A)),groups2906978787729119204at_rat(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7303_sum__strict__mono__ex1,axiom,
    ! [A: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
           => 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),A)
              & 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,A)),groups3906332499630173760nt_rat(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7304_sum__strict__mono__ex1,axiom,
    ! [A: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A)
           => 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),A)
              & 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,A)),groups5058264527183730370ex_rat(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7305_sum__strict__mono__ex1,axiom,
    ! [A: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
           => 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),A)
              & 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,A)),groups4541462559716669496nt_nat(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7306_sum__strict__mono__ex1,axiom,
    ! [A: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A)
           => 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),A)
              & 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,A)),groups5693394587270226106ex_nat(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7307_sum__strict__mono__ex1,axiom,
    ! [A: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(G,X2)) )
       => ( ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A)),groups3539618377306564664at_int(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7308_sum__strict__mono__ex1,axiom,
    ! [A: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X2)),aa_complex_int(G,X2)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X3)),aa_complex_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A)),groups5690904116761175830ex_int(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7309_sum__strict__mono__ex1,axiom,
    ! [A: set_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(G,X2)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X3)),aa_int_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A)),groups4538972089207619220nt_int(G,A)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7310_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X15),Y15)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X2)),aa_nat_complex(G,X2)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_set_nat_complex(groups2073611262835488442omplex(H),S3)),aa_set_nat_complex(groups2073611262835488442omplex(G),S3)) ) ) ) ) ).

% sum.related
tff(fact_7311_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X15),Y15)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_int_complex(H,X2)),aa_int_complex(G,X2)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups3049146728041665814omplex(H,S3)),groups3049146728041665814omplex(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7312_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(plus_plus_real(X15),Y15)),aa_real_real(plus_plus_real(X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => 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,S3)),groups8778361861064173332t_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7313_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(plus_plus_real(X15),Y15)),aa_real_real(plus_plus_real(X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => 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,S3)),groups5808333547571424918x_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7314_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => 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,S3)),groups2906978787729119204at_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7315_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => 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,S3)),groups3906332499630173760nt_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7316_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => 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,S3)),groups5058264527183730370ex_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7317_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => 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,groups4541462559716669496nt_nat(H,S3)),groups4541462559716669496nt_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7318_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => 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,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7319_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X2)),aa_nat_int(G,X2)) )
           => aa_int_o(aa_int_fun_int_o(R,groups3539618377306564664at_int(H,S3)),groups3539618377306564664at_int(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7320_sum__strict__mono,axiom,
    ! [A: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( A != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A)
             => 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,A)),groups5808333547571424918x_real(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7321_sum__strict__mono,axiom,
    ! [A: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => 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,A)),groups8778361861064173332t_real(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7322_sum__strict__mono,axiom,
    ! [A: set_real,F: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A)
             => 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,A)),groups8097168146408367636l_real(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7323_sum__strict__mono,axiom,
    ! [A: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( A != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A)
             => 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,A)),groups5058264527183730370ex_rat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7324_sum__strict__mono,axiom,
    ! [A: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => 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,A)),groups2906978787729119204at_rat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7325_sum__strict__mono,axiom,
    ! [A: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => 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,A)),groups3906332499630173760nt_rat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7326_sum__strict__mono,axiom,
    ! [A: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A)),groups1300246762558778688al_rat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7327_sum__strict__mono,axiom,
    ! [A: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( A != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A)),groups5693394587270226106ex_nat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7328_sum__strict__mono,axiom,
    ! [A: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A)),groups4541462559716669496nt_nat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7329_sum__strict__mono,axiom,
    ! [A: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A)),groups1935376822645274424al_nat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_7330_sum_Oinsert__if,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(Xb,A)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A),groups2240296850493347238T_real(G,A),aa_real_real(plus_plus_real(aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,A))) ) ) ).

% sum.insert_if
tff(fact_7331_sum_Oinsert__if,axiom,
    ! [A: set_real,G: fun_real_real,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( groups8097168146408367636l_real(G,insert_real(Xb,A)) = $ite(aa_set_real_o(member_real(Xb),A),groups8097168146408367636l_real(G,A),aa_real_real(plus_plus_real(aa_real_real(G,Xb)),groups8097168146408367636l_real(G,A))) ) ) ).

% sum.insert_if
tff(fact_7332_sum_Oinsert__if,axiom,
    ! [A: set_int,G: fun_int_real,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups8778361861064173332t_real(G,insert_int(Xb,A)) = $ite(aa_set_int_o(member_int(Xb),A),groups8778361861064173332t_real(G,A),aa_real_real(plus_plus_real(aa_int_real(G,Xb)),groups8778361861064173332t_real(G,A))) ) ) ).

% sum.insert_if
tff(fact_7333_sum_Oinsert__if,axiom,
    ! [A: set_complex,G: fun_complex_real,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5808333547571424918x_real(G,insert_complex(Xb,A)) = $ite(aa_set_complex_o(member_complex(Xb),A),groups5808333547571424918x_real(G,A),aa_real_real(plus_plus_real(aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,A))) ) ) ).

% sum.insert_if
tff(fact_7334_sum_Oinsert__if,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(Xb,A)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A),groups136491112297645522BT_rat(G,A),aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_7335_sum_Oinsert__if,axiom,
    ! [A: set_real,G: fun_real_rat,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( groups1300246762558778688al_rat(G,insert_real(Xb,A)) = $ite(aa_set_real_o(member_real(Xb),A),groups1300246762558778688al_rat(G,A),aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xb)),groups1300246762558778688al_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_7336_sum_Oinsert__if,axiom,
    ! [A: set_nat,G: fun_nat_rat,Xb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( groups2906978787729119204at_rat(G,insert_nat(Xb,A)) = $ite(aa_set_nat_o(member_nat(Xb),A),groups2906978787729119204at_rat(G,A),aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xb)),groups2906978787729119204at_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_7337_sum_Oinsert__if,axiom,
    ! [A: set_int,G: fun_int_rat,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups3906332499630173760nt_rat(G,insert_int(Xb,A)) = $ite(aa_set_int_o(member_int(Xb),A),groups3906332499630173760nt_rat(G,A),aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_7338_sum_Oinsert__if,axiom,
    ! [A: set_complex,G: fun_complex_rat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5058264527183730370ex_rat(G,insert_complex(Xb,A)) = $ite(aa_set_complex_o(member_complex(Xb),A),groups5058264527183730370ex_rat(G,A),aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_7339_sum_Oinsert__if,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(Xb,A)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A),groups771621172384141258BT_nat(G,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,A))) ) ) ).

% sum.insert_if
tff(fact_7340_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
         => ( aa_set_real_o(member_real(I),S)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7341_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S) = zero_zero_real )
         => ( aa_set_int_o(member_int(I),S)
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7342_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(I),S)
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7343_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_rat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,S) = zero_zero_rat )
         => ( aa_set_real_o(member_real(I),S)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7344_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,S) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I),S)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7345_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I),S)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7346_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,S) = zero_zero_rat )
         => ( aa_set_complex_o(member_complex(I),S)
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7347_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_nat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,S) = zero_zero_nat )
         => ( aa_set_real_o(member_real(I),S)
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7348_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S) = zero_zero_nat )
         => ( aa_set_int_o(member_int(I),S)
           => ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7349_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_nat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,S) = zero_zero_nat )
         => ( aa_set_complex_o(member_complex(I),S)
           => ( aa_complex_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7350_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B2: real,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S) = B2 )
         => ( aa_set_real_o(member_real(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7351_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_real,B2: real,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S) = B2 )
         => ( aa_set_int_o(member_int(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7352_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B2: real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S) = B2 )
         => ( aa_set_complex_o(member_complex(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7353_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_rat,B2: rat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,S) = B2 )
         => ( aa_set_real_o(member_real(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7354_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_rat,B2: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,S) = B2 )
         => ( aa_set_nat_o(member_nat(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7355_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_rat,B2: rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S) = B2 )
         => ( aa_set_int_o(member_int(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7356_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_rat,B2: rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,S) = B2 )
         => ( aa_set_complex_o(member_complex(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7357_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_nat,B2: nat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,S) = B2 )
         => ( aa_set_real_o(member_real(I),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7358_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_nat,B2: nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S) = B2 )
         => ( aa_set_int_o(member_int(I),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7359_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_nat,B2: nat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,S) = B2 )
         => ( aa_set_complex_o(member_complex(I),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7360_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A,collect_int(aTP_Lamm_mt(G)))) = groups3049146728041665814omplex(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7361_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A,collect_int(aTP_Lamm_mu(G)))) = groups8778361861064173332t_real(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7362_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A,collect_complex(aTP_Lamm_mv(G)))) = groups5808333547571424918x_real(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7363_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A,collect_int(aTP_Lamm_mw(G)))) = groups3906332499630173760nt_rat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7364_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A,collect_complex(aTP_Lamm_mx(G)))) = groups5058264527183730370ex_rat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7365_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A,collect_int(aTP_Lamm_my(G)))) = groups4541462559716669496nt_nat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7366_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A,collect_complex(aTP_Lamm_mz(G)))) = groups5693394587270226106ex_nat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7367_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A,collect_complex(aTP_Lamm_na(G)))) = groups5690904116761175830ex_int(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7368_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(G),minus_minus_set_nat(A,collect_nat(aTP_Lamm_nb(G)))) = aa_set_nat_complex(groups2073611262835488442omplex(G),A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7369_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( groups2906978787729119204at_rat(G,minus_minus_set_nat(A,collect_nat(aTP_Lamm_nc(G)))) = groups2906978787729119204at_rat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7370_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( aa_set_real_o(member_real(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7371_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7372_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7373_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( aa_set_real_o(member_real(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7374_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7375_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7376_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7377_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( aa_set_real_o(member_real(I),I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I6)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7378_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I6)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7379_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I6)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_7380_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7381_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7382_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7383_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7384_sum__pos,axiom,
    ! [I6: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7385_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7386_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7387_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7388_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7389_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I6)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_7390_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_complex(G,X2) = aa_real_complex(H,X2) ) )
           => ( groups5754745047067104278omplex(G,T2) = groups5754745047067104278omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7391_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7392_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7393_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,T2) = groups1300246762558778688al_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7394_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_rat(G,X2) = aa_complex_rat(H,X2) ) )
           => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7395_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_nat(G,X2) = zero_zero_nat ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_nat(G,X2) = aa_real_nat(H,X2) ) )
           => ( groups1935376822645274424al_nat(G,T2) = groups1935376822645274424al_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7396_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_nat(G,X2) = aa_complex_nat(H,X2) ) )
           => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7397_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_int,H: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_int(G,X2) = zero_zero_int ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_int(G,X2) = aa_real_int(H,X2) ) )
           => ( groups1932886352136224148al_int(G,T2) = groups1932886352136224148al_int(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7398_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_int(G,X2) = aa_complex_int(H,X2) ) )
           => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7399_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => ( aa_nat_complex(G,X2) = aa_nat_complex(H,X2) ) )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(G),T2) = aa_set_nat_complex(groups2073611262835488442omplex(H),S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7400_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_complex(G,X2) = aa_real_complex(H,X2) ) )
           => ( groups5754745047067104278omplex(G,S3) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7401_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7402_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7403_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,S3) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7404_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_rat(G,X2) = aa_complex_rat(H,X2) ) )
           => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7405_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_nat(H,X2) = zero_zero_nat ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_nat(G,X2) = aa_real_nat(H,X2) ) )
           => ( groups1935376822645274424al_nat(G,S3) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7406_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(H,X2) = zero_zero_nat ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_nat(G,X2) = aa_complex_nat(H,X2) ) )
           => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7407_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_int,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T2,S3))
             => ( aa_real_int(H,X2) = zero_zero_int ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_int(G,X2) = aa_real_int(H,X2) ) )
           => ( groups1932886352136224148al_int(G,S3) = groups1932886352136224148al_int(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7408_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(H,X2) = zero_zero_int ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_int(G,X2) = aa_complex_int(H,X2) ) )
           => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7409_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_nat,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => ( aa_nat_complex(G,X2) = aa_nat_complex(H,X2) ) )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(G),S3) = aa_set_nat_complex(groups2073611262835488442omplex(H),T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7410_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7411_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7412_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7413_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7414_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_complex(G,X2) = zero_zero_complex ) )
         => ( aa_set_nat_complex(groups2073611262835488442omplex(G),T2) = aa_set_nat_complex(groups2073611262835488442omplex(G),S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7415_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_rat(G,X2) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T2) = groups2906978787729119204at_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7416_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_int(G,X2) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,T2) = groups3539618377306564664at_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7417_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T2,S3))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7418_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T2,S3))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7419_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T2,S3))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7420_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7421_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7422_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7423_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7424_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_complex(G,X2) = zero_zero_complex ) )
         => ( aa_set_nat_complex(groups2073611262835488442omplex(G),S3) = aa_set_nat_complex(groups2073611262835488442omplex(G),T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7425_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_rat(G,X2) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S3) = groups2906978787729119204at_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7426_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_int(G,X2) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,S3) = groups3539618377306564664at_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7427_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T2,S3))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7428_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T2,S3))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7429_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T2,S3))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7430_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_complex(G,A2) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C2) = groups5754745047067104278omplex(H,C2) )
               => ( groups5754745047067104278omplex(G,A) = groups5754745047067104278omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7431_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_real(G,A2) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) )
               => ( groups8097168146408367636l_real(G,A) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7432_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A))
               => ( aa_complex_real(G,A2) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B2))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) )
               => ( groups5808333547571424918x_real(G,A) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7433_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) )
               => ( groups1300246762558778688al_rat(G,A) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7434_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A))
               => ( aa_complex_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B2))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) )
               => ( groups5058264527183730370ex_rat(G,A) = groups5058264527183730370ex_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7435_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) )
               => ( groups1935376822645274424al_nat(G,A) = groups1935376822645274424al_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7436_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A: set_complex,B2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A))
               => ( aa_complex_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B2))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,C2) = groups5693394587270226106ex_nat(H,C2) )
               => ( groups5693394587270226106ex_nat(G,A) = groups5693394587270226106ex_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7437_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_int,H: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_int(G,A2) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,C2) = groups1932886352136224148al_int(H,C2) )
               => ( groups1932886352136224148al_int(G,A) = groups1932886352136224148al_int(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7438_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A: set_complex,B2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A))
               => ( aa_complex_int(G,A2) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B2))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,C2) = groups5690904116761175830ex_int(H,C2) )
               => ( groups5690904116761175830ex_int(G,A) = groups5690904116761175830ex_int(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7439_sum_Osame__carrierI,axiom,
    ! [C2: set_nat,A: set_nat,B2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),C2)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(member_nat(A2),minus_minus_set_nat(C2,A))
               => ( aa_nat_complex(G,A2) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(C2,B2))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( aa_set_nat_complex(groups2073611262835488442omplex(G),C2) = aa_set_nat_complex(groups2073611262835488442omplex(H),C2) )
               => ( aa_set_nat_complex(groups2073611262835488442omplex(G),A) = aa_set_nat_complex(groups2073611262835488442omplex(H),B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7440_sum_Osame__carrier,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_complex(G,A2) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A) = groups5754745047067104278omplex(H,B2) )
              <=> ( groups5754745047067104278omplex(G,C2) = groups5754745047067104278omplex(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7441_sum_Osame__carrier,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_real(G,A2) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A) = groups8097168146408367636l_real(H,B2) )
              <=> ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7442_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A))
               => ( aa_complex_real(G,A2) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B2))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A) = groups5808333547571424918x_real(H,B2) )
              <=> ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7443_sum_Osame__carrier,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A) = groups1300246762558778688al_rat(H,B2) )
              <=> ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7444_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A))
               => ( aa_complex_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B2))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A) = groups5058264527183730370ex_rat(H,B2) )
              <=> ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7445_sum_Osame__carrier,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A) = groups1935376822645274424al_nat(H,B2) )
              <=> ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7446_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A: set_complex,B2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A))
               => ( aa_complex_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B2))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,A) = groups5693394587270226106ex_nat(H,B2) )
              <=> ( groups5693394587270226106ex_nat(G,C2) = groups5693394587270226106ex_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7447_sum_Osame__carrier,axiom,
    ! [C2: set_real,A: set_real,B2: set_real,G: fun_real_int,H: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C2)
         => ( ! [A2: real] :
                ( aa_set_real_o(member_real(A2),minus_minus_set_real(C2,A))
               => ( aa_real_int(G,A2) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B2))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,A) = groups1932886352136224148al_int(H,B2) )
              <=> ( groups1932886352136224148al_int(G,C2) = groups1932886352136224148al_int(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7448_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A: set_complex,B2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A))
               => ( aa_complex_int(G,A2) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B2))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,A) = groups5690904116761175830ex_int(H,B2) )
              <=> ( groups5690904116761175830ex_int(G,C2) = groups5690904116761175830ex_int(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7449_sum_Osame__carrier,axiom,
    ! [C2: set_nat,A: set_nat,B2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),C2)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(member_nat(A2),minus_minus_set_nat(C2,A))
               => ( aa_nat_complex(G,A2) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(C2,B2))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( aa_set_nat_complex(groups2073611262835488442omplex(G),A) = aa_set_nat_complex(groups2073611262835488442omplex(H),B2) )
              <=> ( aa_set_nat_complex(groups2073611262835488442omplex(G),C2) = aa_set_nat_complex(groups2073611262835488442omplex(H),C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7450_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A)
       => ( groups5808333547571424918x_real(G,A) = aa_real_real(plus_plus_real(groups5808333547571424918x_real(G,minus_811609699411566653omplex(A,B2))),groups5808333547571424918x_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7451_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A)
       => ( groups5058264527183730370ex_rat(G,A) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A,B2))),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7452_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A)
       => ( groups5693394587270226106ex_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A,B2))),groups5693394587270226106ex_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7453_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A)
       => ( groups5690904116761175830ex_int(G,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A,B2))),groups5690904116761175830ex_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7454_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A)
     => ( aa_set_nat_o(finite_finite_nat,A)
       => ( groups2906978787729119204at_rat(G,A) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A,B2))),groups2906978787729119204at_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7455_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A)
     => ( aa_set_nat_o(finite_finite_nat,A)
       => ( groups3539618377306564664at_int(G,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A,B2))),groups3539618377306564664at_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7456_sum_Osubset__diff,axiom,
    ! [B2: set_int,A: set_int,G: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_o(finite_finite_int,A)
       => ( groups8778361861064173332t_real(G,A) = aa_real_real(plus_plus_real(groups8778361861064173332t_real(G,minus_minus_set_int(A,B2))),groups8778361861064173332t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7457_sum_Osubset__diff,axiom,
    ! [B2: set_int,A: set_int,G: fun_int_rat] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_o(finite_finite_int,A)
       => ( groups3906332499630173760nt_rat(G,A) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A,B2))),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7458_sum_Osubset__diff,axiom,
    ! [B2: set_int,A: set_int,G: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_o(finite_finite_int,A)
       => ( groups4541462559716669496nt_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A,B2))),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7459_sum_Osubset__diff,axiom,
    ! [B2: set_int,A: set_int,G: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_o(finite_finite_int,A)
       => ( groups4538972089207619220nt_int(G,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A,B2))),groups4538972089207619220nt_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7460_even__or__iff,axiom,
    ! [Aa: 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))),bit_se1080825931792720795nteger(Aa,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_or_iff
tff(fact_7461_even__or__iff,axiom,
    ! [Aa: 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(bit_se1409905431419307370or_int,Aa),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_or_iff
tff(fact_7462_even__or__iff,axiom,
    ! [Aa: 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(bit_se1412395901928357646or_nat,Aa),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_or_iff
tff(fact_7463_sum__diff,axiom,
    ! [A: set_complex,B2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A)
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A,B2)) = aa_real_real(minus_minus_real(groups5808333547571424918x_real(F,A)),groups5808333547571424918x_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7464_sum__diff,axiom,
    ! [A: set_complex,B2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A)
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A,B2)) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7465_sum__diff,axiom,
    ! [A: set_complex,B2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A)
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A)),groups5690904116761175830ex_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7466_sum__diff,axiom,
    ! [A: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),A)
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A,B2)) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A)),groups2906978787729119204at_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7467_sum__diff,axiom,
    ! [A: set_nat,B2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),A)
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A)),groups3539618377306564664at_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7468_sum__diff,axiom,
    ! [A: set_int,B2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A)
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A,B2)) = aa_real_real(minus_minus_real(groups8778361861064173332t_real(F,A)),groups8778361861064173332t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7469_sum__diff,axiom,
    ! [A: set_int,B2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A)
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A,B2)) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A)),groups3906332499630173760nt_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7470_sum__diff,axiom,
    ! [A: set_int,B2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A)
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A)),groups4538972089207619220nt_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7471_sum__diff,axiom,
    ! [A: set_complex,B2: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A)
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A,B2)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups7754918857620584856omplex(F,A)),groups7754918857620584856omplex(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7472_sum__diff,axiom,
    ! [A: set_nat,B2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),A)
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),minus_minus_set_nat(A,B2)) = aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(F),A)),aa_set_nat_real(groups6591440286371151544t_real(F),B2)) ) ) ) ).

% sum_diff
tff(fact_7473_bit_Ocomplement__unique,axiom,
    ! [Aa: code_integer,Xb: code_integer,Y3: code_integer] :
      ( ( bit_se3949692690581998587nteger(Aa,Xb) = zero_z3403309356797280102nteger )
     => ( ( bit_se1080825931792720795nteger(Aa,Xb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
       => ( ( bit_se3949692690581998587nteger(Aa,Y3) = zero_z3403309356797280102nteger )
         => ( ( bit_se1080825931792720795nteger(Aa,Y3) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( Xb = Y3 ) ) ) ) ) ).

% bit.complement_unique
tff(fact_7474_bit_Ocomplement__unique,axiom,
    ! [Aa: int,Xb: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),Xb) = zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),Xb) = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Aa),Y3) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),Y3) = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( Xb = Y3 ) ) ) ) ) ).

% bit.complement_unique
tff(fact_7475_sum__mono2,axiom,
    ! [B2: set_real,A: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),B2)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B2,A))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A)),groups8097168146408367636l_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7476_sum__mono2,axiom,
    ! [B2: set_complex,A: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B2,A))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A)),groups5808333547571424918x_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7477_sum__mono2,axiom,
    ! [B2: set_real,A: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),B2)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B2,A))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A)),groups1300246762558778688al_rat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7478_sum__mono2,axiom,
    ! [B2: set_complex,A: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B2,A))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7479_sum__mono2,axiom,
    ! [B2: set_nat,A: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B2,A))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A)),groups2906978787729119204at_rat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7480_sum__mono2,axiom,
    ! [B2: set_real,A: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),B2)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B2,A))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A)),groups1935376822645274424al_nat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7481_sum__mono2,axiom,
    ! [B2: set_complex,A: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B2,A))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7482_sum__mono2,axiom,
    ! [B2: set_real,A: set_real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),B2)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B2,A))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A)),groups1932886352136224148al_int(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7483_sum__mono2,axiom,
    ! [B2: set_complex,A: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B2,A))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,A)),groups5690904116761175830ex_int(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7484_sum__mono2,axiom,
    ! [B2: set_nat,A: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B2,A))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,A)),groups3539618377306564664at_int(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7485_sum_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(Xb,A)) = aa_real_real(plus_plus_real(aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7486_sum_Oinsert__remove,axiom,
    ! [A: set_complex,G: fun_complex_real,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5808333547571424918x_real(G,insert_complex(Xb,A)) = aa_real_real(plus_plus_real(aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7487_sum_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(Xb,A)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7488_sum_Oinsert__remove,axiom,
    ! [A: set_complex,G: fun_complex_rat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5058264527183730370ex_rat(G,insert_complex(Xb,A)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7489_sum_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(Xb,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7490_sum_Oinsert__remove,axiom,
    ! [A: set_complex,G: fun_complex_nat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5693394587270226106ex_nat(G,insert_complex(Xb,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7491_sum_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,Xb: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( groups769130701875090982BT_int(G,insert_VEBT_VEBT(Xb,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,Xb)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7492_sum_Oinsert__remove,axiom,
    ! [A: set_complex,G: fun_complex_int,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( groups5690904116761175830ex_int(G,insert_complex(Xb,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,Xb)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7493_sum_Oinsert__remove,axiom,
    ! [A: set_int,G: fun_int_real,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups8778361861064173332t_real(G,insert_int(Xb,A)) = aa_real_real(plus_plus_real(aa_int_real(G,Xb)),groups8778361861064173332t_real(G,minus_minus_set_int(A,insert_int(Xb,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_7494_sum_Oinsert__remove,axiom,
    ! [A: set_int,G: fun_int_rat,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups3906332499630173760nt_rat(G,insert_int(Xb,A)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A,insert_int(Xb,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_7495_sum_Oremove,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A)
       => ( groups2240296850493347238T_real(G,A) = aa_real_real(plus_plus_real(aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7496_sum_Oremove,axiom,
    ! [A: set_complex,Xb: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_complex_o(member_complex(Xb),A)
       => ( groups5808333547571424918x_real(G,A) = aa_real_real(plus_plus_real(aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7497_sum_Oremove,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A)
       => ( groups136491112297645522BT_rat(G,A) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7498_sum_Oremove,axiom,
    ! [A: set_complex,Xb: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_complex_o(member_complex(Xb),A)
       => ( groups5058264527183730370ex_rat(G,A) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7499_sum_Oremove,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A)
       => ( groups771621172384141258BT_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7500_sum_Oremove,axiom,
    ! [A: set_complex,Xb: complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_complex_o(member_complex(Xb),A)
       => ( groups5693394587270226106ex_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7501_sum_Oremove,axiom,
    ! [A: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A)
       => ( groups769130701875090982BT_int(G,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,Xb)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7502_sum_Oremove,axiom,
    ! [A: set_complex,Xb: complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_complex_o(member_complex(Xb),A)
       => ( groups5690904116761175830ex_int(G,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,Xb)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7503_sum_Oremove,axiom,
    ! [A: set_int,Xb: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(Xb),A)
       => ( groups8778361861064173332t_real(G,A) = aa_real_real(plus_plus_real(aa_int_real(G,Xb)),groups8778361861064173332t_real(G,minus_minus_set_int(A,insert_int(Xb,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_7504_sum_Oremove,axiom,
    ! [A: set_int,Xb: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(Xb),A)
       => ( groups3906332499630173760nt_rat(G,A) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A,insert_int(Xb,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_7505_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,Aa: vEBT_VEBT,B: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_nd(Aa),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Aa),S3),aa_real_real(plus_plus_real(aa_VEBT_VEBT_real(B,Aa)),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT)))),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_7506_sum_Odelta__remove,axiom,
    ! [S3: set_complex,Aa: complex,B: fun_complex_real,C: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_ne(Aa),B),C),S3) = $ite(aa_set_complex_o(member_complex(Aa),S3),aa_real_real(plus_plus_real(aa_complex_real(B,Aa)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S3,insert_complex(Aa,bot_bot_set_complex)))),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S3,insert_complex(Aa,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_7507_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,Aa: vEBT_VEBT,B: fun_VEBT_VEBT_rat,C: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_nf(Aa),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Aa),S3),aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(B,Aa)),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT)))),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_7508_sum_Odelta__remove,axiom,
    ! [S3: set_complex,Aa: complex,B: fun_complex_rat,C: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_ng(Aa),B),C),S3) = $ite(aa_set_complex_o(member_complex(Aa),S3),aa_rat_rat(plus_plus_rat(aa_complex_rat(B,Aa)),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(Aa,bot_bot_set_complex)))),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(Aa,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_7509_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,Aa: vEBT_VEBT,B: fun_VEBT_VEBT_nat,C: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_nh(Aa),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Aa),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(B,Aa)),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT)))),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_7510_sum_Odelta__remove,axiom,
    ! [S3: set_complex,Aa: complex,B: fun_complex_nat,C: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ni(Aa),B),C),S3) = $ite(aa_set_complex_o(member_complex(Aa),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,Aa)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(Aa,bot_bot_set_complex)))),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(Aa,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_7511_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,Aa: vEBT_VEBT,B: fun_VEBT_VEBT_int,C: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_nj(Aa),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Aa),S3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(B,Aa)),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT)))),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Aa,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_7512_sum_Odelta__remove,axiom,
    ! [S3: set_complex,Aa: complex,B: fun_complex_int,C: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_nk(Aa),B),C),S3) = $ite(aa_set_complex_o(member_complex(Aa),S3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B,Aa)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S3,insert_complex(Aa,bot_bot_set_complex)))),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S3,insert_complex(Aa,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_7513_sum_Odelta__remove,axiom,
    ! [S3: set_int,Aa: int,B: fun_int_real,C: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_nl(Aa),B),C),S3) = $ite(aa_set_int_o(member_int(Aa),S3),aa_real_real(plus_plus_real(aa_int_real(B,Aa)),groups8778361861064173332t_real(C,minus_minus_set_int(S3,insert_int(Aa,bot_bot_set_int)))),groups8778361861064173332t_real(C,minus_minus_set_int(S3,insert_int(Aa,bot_bot_set_int)))) ) ) ).

% sum.delta_remove
tff(fact_7514_sum_Odelta__remove,axiom,
    ! [S3: set_int,Aa: int,B: fun_int_rat,C: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_nm(Aa),B),C),S3) = $ite(aa_set_int_o(member_int(Aa),S3),aa_rat_rat(plus_plus_rat(aa_int_rat(B,Aa)),groups3906332499630173760nt_rat(C,minus_minus_set_int(S3,insert_int(Aa,bot_bot_set_int)))),groups3906332499630173760nt_rat(C,minus_minus_set_int(S3,insert_int(Aa,bot_bot_set_int)))) ) ) ).

% sum.delta_remove
tff(fact_7515_sum__strict__mono2,axiom,
    ! [B2: set_real,A: set_real,B: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A))
         => ( 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),B2)
                 => 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,A)),groups8097168146408367636l_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7516_sum__strict__mono2,axiom,
    ! [B2: set_complex,A: set_complex,B: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A))
         => ( 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),B2)
                 => 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,A)),groups5808333547571424918x_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7517_sum__strict__mono2,axiom,
    ! [B2: set_real,A: set_real,B: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A))
         => ( 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),B2)
                 => 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,A)),groups1300246762558778688al_rat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7518_sum__strict__mono2,axiom,
    ! [B2: set_complex,A: set_complex,B: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A))
         => ( 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),B2)
                 => 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,A)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7519_sum__strict__mono2,axiom,
    ! [B2: set_nat,A: set_nat,B: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B2,A))
         => ( 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),B2)
                 => 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,A)),groups2906978787729119204at_rat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7520_sum__strict__mono2,axiom,
    ! [B2: set_real,A: set_real,B: real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,B))
           => ( ! [X2: real] :
                  ( aa_set_real_o(member_real(X2),B2)
                 => 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_nat,groups1935376822645274424al_nat(F,A)),groups1935376822645274424al_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7521_sum__strict__mono2,axiom,
    ! [B2: set_complex,A: set_complex,B: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B))
           => ( ! [X2: complex] :
                  ( aa_set_complex_o(member_complex(X2),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X2)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7522_sum__strict__mono2,axiom,
    ! [B2: set_real,A: set_real,B: real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(F,B))
           => ( ! [X2: real] :
                  ( aa_set_real_o(member_real(X2),B2)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X2)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups1932886352136224148al_int(F,A)),groups1932886352136224148al_int(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7523_sum__strict__mono2,axiom,
    ! [B2: set_complex,A: set_complex,B: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_complex_int(F,B))
           => ( ! [X2: complex] :
                  ( aa_set_complex_o(member_complex(X2),B2)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X2)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A)),groups5690904116761175830ex_int(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7524_sum__strict__mono2,axiom,
    ! [B2: set_nat,A: set_nat,B: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B2)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B2,A))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,B))
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(member_nat(X2),B2)
                 => 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_int,groups3539618377306564664at_int(F,A)),groups3539618377306564664at_int(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7525_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(A,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,A)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),groups2240296850493347238T_real(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7526_member__le__sum,axiom,
    ! [I: complex,A: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(member_complex(I),A)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(A,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,A)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7527_member__le__sum,axiom,
    ! [I: int,A: set_int,F: fun_int_real] :
      ( aa_set_int_o(member_int(I),A)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),minus_minus_set_int(A,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,A)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7528_member__le__sum,axiom,
    ! [I: real,A: set_real,F: fun_real_real] :
      ( aa_set_real_o(member_real(I),A)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),minus_minus_set_real(A,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,A)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7529_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(A,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,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),groups136491112297645522BT_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7530_member__le__sum,axiom,
    ! [I: complex,A: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(member_complex(I),A)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(A,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,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7531_member__le__sum,axiom,
    ! [I: int,A: set_int,F: fun_int_rat] :
      ( aa_set_int_o(member_int(I),A)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),minus_minus_set_int(A,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,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7532_member__le__sum,axiom,
    ! [I: real,A: set_real,F: fun_real_rat] :
      ( aa_set_real_o(member_real(I),A)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),minus_minus_set_real(A,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,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7533_member__le__sum,axiom,
    ! [I: nat,A: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(member_nat(I),A)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(A,insert_nat(I,bot_bot_set_nat)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
       => ( aa_set_nat_o(finite_finite_nat,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),groups2906978787729119204at_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7534_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X2)) )
       => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I)),groups771621172384141258BT_nat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_7535_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_co8153928105989009288nteger,Aa: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(Xb,I2)) )
     => ( ( groups6621422865394947399nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(Aa,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_nn(Xb),Aa),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7536_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_re6978948838062957830nteger,Aa: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(Xb,I2)) )
     => ( ( groups7713935264441627589nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(Aa,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_no(Xb),Aa),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7537_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_Code_integer,Aa: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Xb,I2)) )
     => ( ( groups7501900531339628137nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(Aa,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_np(Xb),Aa),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7538_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_Code_integer,Aa: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Xb,I2)) )
     => ( ( groups7873554091576472773nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(Aa,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_nq(Xb),Aa),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7539_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,Aa: fun_complex_real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Xb,I2)) )
     => ( ( groups5808333547571424918x_real(Xb,I6) = one_one_real )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_complex_real(Aa,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_nr(Xb),Aa),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7540_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_real,Aa: fun_real_real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Xb,I2)) )
     => ( ( groups8097168146408367636l_real(Xb,I6) = one_one_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(Aa,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ns(Xb),Aa),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7541_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_real,Aa: fun_int_real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Xb,I2)) )
     => ( ( groups8778361861064173332t_real(Xb,I6) = one_one_real )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(Aa,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_nt(Xb),Aa),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7542_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_complex_rat,Aa: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Xb,I2)) )
     => ( ( groups5058264527183730370ex_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_complex_rat(Aa,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_nu(Xb),Aa),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7543_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_rat,Aa: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Xb,I2)) )
     => ( ( groups1300246762558778688al_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(Aa,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nv(Xb),Aa),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7544_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,Aa: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Xb,I2)) )
     => ( ( groups2906978787729119204at_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(Aa,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_nw(Xb),Aa),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7545_one__or__eq,axiom,
    ! [Aa: code_integer] : bit_se1080825931792720795nteger(one_one_Code_integer,Aa) = plus_p5714425477246183910nteger(Aa,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa))) ).

% one_or_eq
tff(fact_7546_one__or__eq,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),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))),Aa))) ).

% one_or_eq
tff(fact_7547_one__or__eq,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),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))),Aa))) ).

% one_or_eq
tff(fact_7548_or__one__eq,axiom,
    ! [Aa: code_integer] : bit_se1080825931792720795nteger(Aa,one_one_Code_integer) = plus_p5714425477246183910nteger(Aa,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa))) ).

% or_one_eq
tff(fact_7549_or__one__eq,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Aa),one_one_int) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),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))),Aa))) ).

% or_one_eq
tff(fact_7550_or__one__eq,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Aa),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),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))),Aa))) ).

% or_one_eq
tff(fact_7551_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Y3: 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,Y3),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),Y3)),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_7552_add__0__iff,axiom,
    ! [B: complex,Aa: complex] :
      ( ( B = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),Aa) )
    <=> ( Aa = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_7553_add__0__iff,axiom,
    ! [B: real,Aa: real] :
      ( ( B = aa_real_real(plus_plus_real(B),Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

% add_0_iff
tff(fact_7554_add__0__iff,axiom,
    ! [B: rat,Aa: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(B),Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_7555_add__0__iff,axiom,
    ! [B: nat,Aa: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Aa) )
    <=> ( Aa = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_7556_add__0__iff,axiom,
    ! [B: int,Aa: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Aa) )
    <=> ( Aa = zero_zero_int ) ) ).

% add_0_iff
tff(fact_7557_crossproduct__eq,axiom,
    ! [W: real,Y3: real,Xb: real,Z: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(W),Y3)),aa_real_real(times_times_real(Xb),Z)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(W),Z)),aa_real_real(times_times_real(Xb),Y3)) )
    <=> ( ( W = Xb )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
tff(fact_7558_crossproduct__eq,axiom,
    ! [W: rat,Y3: rat,Xb: rat,Z: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Y3)),aa_rat_rat(times_times_rat(Xb),Z)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Z)),aa_rat_rat(times_times_rat(Xb),Y3)) )
    <=> ( ( W = Xb )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
tff(fact_7559_crossproduct__eq,axiom,
    ! [W: nat,Y3: nat,Xb: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y3)) )
    <=> ( ( W = Xb )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
tff(fact_7560_crossproduct__eq,axiom,
    ! [W: int,Y3: int,Xb: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y3)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y3)) )
    <=> ( ( W = Xb )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
tff(fact_7561_crossproduct__noteq,axiom,
    ! [Aa: real,B: real,C: real,D: real] :
      ( ( ( Aa != B )
        & ( C != D ) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(B),D)) != aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),D)),aa_real_real(times_times_real(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_7562_crossproduct__noteq,axiom,
    ! [Aa: rat,B: rat,C: rat,D: rat] :
      ( ( ( Aa != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),D)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_7563_crossproduct__noteq,axiom,
    ! [Aa: nat,B: nat,C: nat,D: nat] :
      ( ( ( Aa != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_7564_crossproduct__noteq,axiom,
    ! [Aa: int,B: int,C: int,D: int] :
      ( ( ( Aa != B )
        & ( C != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_7565_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_7566_accp__subset__induct,axiom,
    ! [D2: fun_Pr7546709192626087934_nat_o,R: fun_Pr8877776288780692068_nat_o,Xb: produc9072475918466114483BT_nat,P: fun_Pr7546709192626087934_nat_o] :
      ( ord_le1769015658490769310_nat_o(D2,accp_P2887432264394892906BT_nat(R))
     => ( ! [X2: produc9072475918466114483BT_nat,Z3: produc9072475918466114483BT_nat] :
            ( aa_Pro632845113475883111_nat_o(D2,X2)
           => ( aa_Pro632845113475883111_nat_o(aa_Pro1578556086376436173_nat_o(R,Z3),X2)
             => aa_Pro632845113475883111_nat_o(D2,Z3) ) )
       => ( aa_Pro632845113475883111_nat_o(D2,Xb)
         => ( ! [X2: produc9072475918466114483BT_nat] :
                ( aa_Pro632845113475883111_nat_o(D2,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_7567_accp__subset__induct,axiom,
    ! [D2: fun_Pr6331313853612757382_num_o,R: fun_Pr4799464687642144302_num_o,Xb: product_prod_num_num,P: fun_Pr6331313853612757382_num_o] :
      ( ord_le3107050023622517222_num_o(D2,accp_P3113834385874906142um_num(R))
     => ( ! [X2: product_prod_num_num,Z3: product_prod_num_num] :
            ( aa_Pro7685007914727650815_num_o(D2,X2)
           => ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(R,Z3),X2)
             => aa_Pro7685007914727650815_num_o(D2,Z3) ) )
       => ( aa_Pro7685007914727650815_num_o(D2,Xb)
         => ( ! [X2: product_prod_num_num] :
                ( aa_Pro7685007914727650815_num_o(D2,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_7568_accp__subset__induct,axiom,
    ! [D2: fun_Pr4796943543331071514_nat_o,R: fun_Pr2028404973800837934_nat_o,Xb: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( ord_le1572679713340831354_nat_o(D2,accp_P4275260045618599050at_nat(R))
     => ( ! [X2: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(D2,X2)
           => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Z3),X2)
             => aa_Pro6150637604445964947_nat_o(D2,Z3) ) )
       => ( aa_Pro6150637604445964947_nat_o(D2,Xb)
         => ( ! [X2: product_prod_nat_nat] :
                ( aa_Pro6150637604445964947_nat_o(D2,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_7569_accp__subset__induct,axiom,
    ! [D2: fun_Pr3238374608701176162_int_o,R: fun_Pr9018117161326028078_int_o,Xb: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( ord_le14110778710936002_int_o(D2,accp_P1096762738010456898nt_int(R))
     => ( ! [X2: product_prod_int_int,Z3: product_prod_int_int] :
            ( aa_Pro4592068669816069595_int_o(D2,X2)
           => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R,Z3),X2)
             => aa_Pro4592068669816069595_int_o(D2,Z3) ) )
       => ( aa_Pro4592068669816069595_int_o(D2,Xb)
         => ( ! [X2: product_prod_int_int] :
                ( aa_Pro4592068669816069595_int_o(D2,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_7570_accp__subset__induct,axiom,
    ! [D2: fun_nat_o,R: fun_nat_fun_nat_o,Xb: nat,P: fun_nat_o] :
      ( ord_le7240017405068453697_nat_o(D2,accp_nat(R))
     => ( ! [X2: nat,Z3: nat] :
            ( aa_nat_o(D2,X2)
           => ( aa_nat_o(aa_nat_fun_nat_o(R,Z3),X2)
             => aa_nat_o(D2,Z3) ) )
       => ( aa_nat_o(D2,Xb)
         => ( ! [X2: nat] :
                ( aa_nat_o(D2,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_7571_arctan__add,axiom,
    ! [Xb: real,Y3: 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,Y3)),one_one_real)
       => ( aa_real_real(plus_plus_real(aa_real_real(arctan,Xb)),aa_real_real(arctan,Y3)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Y3)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(Xb),Y3)))) ) ) ) ).

% arctan_add
tff(fact_7572_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y3 )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xb)
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = $ite(
                      (B3),
                      some_nat(one_one_nat),
                      $ite((A2),some_nat(zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A2),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y3 = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y3 = some_nat(Ma) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),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_7573_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y3 )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xb)
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = $ite(
                      (A2),
                      some_nat(zero_zero_nat),
                      $ite((B3),some_nat(one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A2),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y3 = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y3 = some_nat(Mi) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),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_7574_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xb) = Y3 )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),Xb)
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((A2),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((A2),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y3 = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),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_7575_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xb) = Y3 )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),Xb)
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((B3),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((A2),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y3 = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),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_7576_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_num_num,A1: nat,A22: nat,A32: num,P: fun_fu8148986511470497476_num_o] :
      ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(A0,produc1195630363706982562at_num(A1,product_Pair_nat_num(A22,A32))))
     => ( ! [F2: fun_nat_fun_num_num,A2: nat,B3: nat,Acc: num] :
            ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(F2,produc1195630363706982562at_num(A2,product_Pair_nat_num(B3,Acc))))
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),A2)
               => aa_num_o(aa_nat_fun_num_o(aa_nat9072979527818120146_num_o(aa_fun2776124910339071917_num_o(P,F2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B3),aa_num_num(aa_nat_fun_num_num(F2,A2),Acc)) )
             => aa_num_o(aa_nat_fun_num_o(aa_nat9072979527818120146_num_o(aa_fun2776124910339071917_num_o(P,F2),A2),B3),Acc) ) )
       => aa_num_o(aa_nat_fun_num_o(aa_nat9072979527818120146_num_o(aa_fun2776124910339071917_num_o(P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_7577_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_nat_nat,A1: nat,A22: nat,A32: nat,P: fun_fu2571895429041709858_nat_o] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(A0,produc487386426758144856at_nat(A1,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A22),A32))))
     => ( ! [F2: fun_nat_fun_nat_nat,A2: nat,B3: nat,Acc: nat] :
            ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A2,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B3),Acc))))
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),A2)
               => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,F2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B3),aa_nat_nat(aa_nat_fun_nat_nat(F2,A2),Acc)) )
             => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,F2),A2),B3),Acc) ) )
       => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_7578_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y3 )
     => ( accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_P5556105721700978146at_nat )
           => ( ( Y3 = 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 )
                 => ( ( Y3 = none_P5556105721700978146at_nat )
                   => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))) ) ) )
           => ~ ! [A2: product_prod_nat_nat] :
                  ( ( Xaa = some_P7363390416028606310at_nat(A2) )
                 => ! [B3: product_prod_nat_nat] :
                      ( ( Xba = some_P7363390416028606310at_nat(B3) )
                     => ( ( Y3 = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A2),B3)) )
                       => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A2),some_P7363390416028606310at_nat(B3)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7579_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y3: option_num] :
      ( ( vEBT_V819420779217536731ft_num(Xb,Xaa,Xba) = Y3 )
     => ( accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(Xaa,Xba)))
       => ( ( ( Xaa = none_num )
           => ( ( Y3 = none_num )
             => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(none_num,Xba))) ) )
         => ( ! [V2: num] :
                ( ( Xaa = some_num(V2) )
               => ( ( Xba = none_num )
                 => ( ( Y3 = none_num )
                   => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(some_num(V2),none_num))) ) ) )
           => ~ ! [A2: num] :
                  ( ( Xaa = some_num(A2) )
                 => ! [B3: num] :
                      ( ( Xba = some_num(B3) )
                     => ( ( Y3 = some_num(aa_num_num(aa_num_fun_num_num(Xb,A2),B3)) )
                       => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(some_num(A2),some_num(B3)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7580_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y3: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y3 )
     => ( accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_nat )
           => ( ( Y3 = none_nat )
             => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(none_nat,Xba))) ) )
         => ( ! [V2: nat] :
                ( ( Xaa = some_nat(V2) )
               => ( ( Xba = none_nat )
                 => ( ( Y3 = none_nat )
                   => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(some_nat(V2),none_nat))) ) ) )
           => ~ ! [A2: nat] :
                  ( ( Xaa = some_nat(A2) )
                 => ! [B3: nat] :
                      ( ( Xba = some_nat(B3) )
                     => ( ( Y3 = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xb,A2),B3)) )
                       => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(some_nat(A2),some_nat(B3)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7581_or__nat__numerals_I2_J,axiom,
    ! [Y3: 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,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_nat_numerals(2)
tff(fact_7582_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_7583_or__nat__numerals_I1_J,axiom,
    ! [Y3: 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,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_nat_numerals(1)
tff(fact_7584_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_7585_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_set_nat_complex(groups2073611262835488442omplex(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7586_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7587_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7588_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7589_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_real,aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7590_sum__zero__power,axiom,
    ! [C: fun_nat_complex,A: set_nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_nx(C)),A) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A)
        & aa_set_nat_o(member_nat(zero_zero_nat),A) ),
        aa_nat_complex(C,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_7591_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_ny(C),A) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A)
        & aa_set_nat_o(member_nat(zero_zero_nat),A) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_7592_sum__zero__power,axiom,
    ! [C: fun_nat_real,A: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nz(C)),A) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A)
        & aa_set_nat_o(member_nat(zero_zero_nat),A) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_7593_sum__zero__power_H,axiom,
    ! [C: fun_nat_complex,D: fun_nat_complex,A: set_nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_oa(C),D)),A) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A)
        & aa_set_nat_o(member_nat(zero_zero_nat),A) ),
        aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)),
        zero_zero_complex ) ).

% sum_zero_power'
tff(fact_7594_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ob(C),D),A) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A)
        & aa_set_nat_o(member_nat(zero_zero_nat),A) ),
        aa_rat_rat(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_7595_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_oc(C),D)),A) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A)
        & aa_set_nat_o(member_nat(zero_zero_nat),A) ),
        aa_real_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_7596_sum__cong__Suc,axiom,
    ! [A: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A)
           => ( 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),A) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A) ) ) ) ).

% sum_cong_Suc
tff(fact_7597_sum__cong__Suc,axiom,
    ! [A: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A)
           => ( aa_nat_real(F,aa_nat_nat(suc,X2)) = aa_nat_real(G,aa_nat_nat(suc,X2)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A) = aa_set_nat_real(groups6591440286371151544t_real(G),A) ) ) ) ).

% sum_cong_Suc
tff(fact_7598_sum__subtractf__nat,axiom,
    ! [A: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(G,X2)),aa_complex_nat(F,X2)) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_od(G),F),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A)),groups5693394587270226106ex_nat(G,A)) ) ) ).

% sum_subtractf_nat
tff(fact_7599_sum__subtractf__nat,axiom,
    ! [A: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A)
         => 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_oe(G),F),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A)),groups1935376822645274424al_nat(G,A)) ) ) ).

% sum_subtractf_nat
tff(fact_7600_sum__subtractf__nat,axiom,
    ! [A: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X2)),aa_set_nat_nat(F,X2)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_of(G),F),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A)),groups8294997508430121362at_nat(G,A)) ) ) ).

% sum_subtractf_nat
tff(fact_7601_sum__subtractf__nat,axiom,
    ! [A: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => 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_og(G),F),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A)),groups4541462559716669496nt_nat(G,A)) ) ) ).

% sum_subtractf_nat
tff(fact_7602_sum__subtractf__nat,axiom,
    ! [A: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => 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_oh(G),F)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A)) ) ) ).

% sum_subtractf_nat
tff(fact_7603_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_oi(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7604_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oj(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7605_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: 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,Mb),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_ok(G),K)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7606_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,K: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ol(G),K)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7607_sum__eq__Suc0__iff,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( groups4541462559716669496nt_nat(F,A) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X: int] :
            ( aa_set_int_o(member_int(X),A)
            & ( aa_int_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A)
               => ( ( X != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7608_sum__eq__Suc0__iff,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( groups5693394587270226106ex_nat(F,A) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A)
            & ( aa_complex_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A)
               => ( ( X != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7609_sum__eq__Suc0__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A)
            & ( aa_nat_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A)
               => ( ( X != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7610_sum__SucD,axiom,
    ! [F: fun_nat_nat,A: set_nat,Nb: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A) = aa_nat_nat(suc,Nb) )
     => ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X2)) ) ) ).

% sum_SucD
tff(fact_7611_sum__eq__1__iff,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( groups4541462559716669496nt_nat(F,A) = one_one_nat )
      <=> ? [X: int] :
            ( aa_set_int_o(member_int(X),A)
            & ( aa_int_nat(F,X) = one_one_nat )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A)
               => ( ( X != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7612_sum__eq__1__iff,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( groups5693394587270226106ex_nat(F,A) = one_one_nat )
      <=> ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A)
            & ( aa_complex_nat(F,X) = one_one_nat )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A)
               => ( ( X != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7613_sum__eq__1__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A) = one_one_nat )
      <=> ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A)
            & ( aa_nat_nat(F,X) = one_one_nat )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A)
               => ( ( X != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7614_sum__power__add,axiom,
    ! [Xb: complex,Mb: nat,I6: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_om(Xb),Mb)),I6) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Mb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),I6)) ).

% sum_power_add
tff(fact_7615_sum__power__add,axiom,
    ! [Xb: rat,Mb: nat,I6: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_on(Xb),Mb),I6) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),I6)) ).

% sum_power_add
tff(fact_7616_sum__power__add,axiom,
    ! [Xb: int,Mb: nat,I6: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_oo(Xb),Mb),I6) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Mb)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),I6)) ).

% sum_power_add
tff(fact_7617_sum__power__add,axiom,
    ! [Xb: real,Mb: nat,I6: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_op(Xb),Mb)),I6) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),I6)) ).

% sum_power_add
tff(fact_7618_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Nb,Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oq(G),Nb),Mb)),set_or1269000886237332187st_nat(Nb,Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_7619_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Nb,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_or(G),Nb),Mb)),set_or1269000886237332187st_nat(Nb,Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_7620_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_af,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_en(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_7621_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_af,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_do,Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_7622_sum__diff__nat,axiom,
    ! [B2: set_complex,A: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A)
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_7623_sum__diff__nat,axiom,
    ! [B2: set_int,A: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A)
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A)),groups4541462559716669496nt_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_7624_sum__diff__nat,axiom,
    ! [B2: set_nat,A: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),A)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B2)) ) ) ) ).

% sum_diff_nat
tff(fact_7625_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7626_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_7627_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_7628_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_7629_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7630_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_7631_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_7632_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_7633_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7634_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7635_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7636_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7637_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,aa_nat_nat(suc,Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7638_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7639_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7640_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7641_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb)) = aa_real_real(plus_plus_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7642_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(aTP_Lamm_os(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7643_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(aTP_Lamm_ot(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7644_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_oi(G)),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7645_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oj(G)),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7646_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(aTP_Lamm_ou(F),set_or1269000886237332187st_nat(Mb,Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Nb))),aa_nat_rat(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_7647_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(aTP_Lamm_ov(F),set_or1269000886237332187st_nat(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_7648_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ow(F)),set_or1269000886237332187st_nat(Mb,Nb)) = aa_real_real(minus_minus_real(aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_7649_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7650_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7651_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7652_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7653_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_7654_sum__natinterval__diff,axiom,
    ! [F: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ox(F)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(F,Mb)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_complex) ).

% sum_natinterval_diff
tff(fact_7655_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_oy(F),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_7656_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(aTP_Lamm_oz(F),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_7657_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pa(F)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_real_real(minus_minus_real(aa_nat_real(F,Mb)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_7658_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_pb(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7659_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_pc(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7660_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pd(F)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(F,Nb)),aa_nat_real(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7661_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_dj,Nb))) ).

% mask_eq_sum_exp
tff(fact_7662_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_dj,Nb))) ).

% mask_eq_sum_exp
tff(fact_7663_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),set_or1269000886237332187st_nat(Mb,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7664_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,Nb))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7665_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),set_or1269000886237332187st_nat(Mb,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7666_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,Nb))) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7667_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_pe(G),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_7668_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_pf(G),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_7669_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pg(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_7670_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ph(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_7671_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_7672_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_7673_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          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))),Mb)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_7674_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_dj,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_7675_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),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_7676_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_7677_arith__series__nat,axiom,
    ! [Aa: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pi(Aa),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))),Aa)),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_7678_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_7679_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,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xb) = Y3 )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( ( Y3 = 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)) )
               => ( ( Y3 = 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) )
                 => ( ( Y3 = 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 = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ( Y3 = one_one_nat )
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_7680_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [Xb: fun_nat_fun_num_num,Xaa: nat,Xba: nat,Xc: num,Y3: num] :
      ( ( aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(Xb,Xaa),Xba),Xc) = Y3 )
     => ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(Xb,produc1195630363706982562at_num(Xaa,product_Pair_nat_num(Xba,Xc))))
       => ~ ( ( Y3 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_num_num(aa_nat_fun_num_num(Xb,Xaa),Xc))) )
           => ~ accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(Xb,produc1195630363706982562at_num(Xaa,product_Pair_nat_num(Xba,Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_7681_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y3 )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xb,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xba),Xc))))
       => ~ ( ( Y3 = $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_7682_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_num_num,Aa: nat,B: nat,Acc2: num] :
      ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(F,produc1195630363706982562at_num(Aa,product_Pair_nat_num(B,Acc2))))
     => ( aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(F,Aa),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa),Acc2,aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),B),aa_num_num(aa_nat_fun_num_num(F,Aa),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_7683_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_nat_nat,Aa: nat,B: nat,Acc2: nat] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F,produc487386426758144856at_nat(Aa,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Acc2))))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,Aa),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,Aa),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_7684_in__measure,axiom,
    ! [Xb: num,Y3: num,F: fun_num_nat] :
      ( member7279096912039735102um_num(product_Pair_num_num(Xb,Y3),measure_num(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,Xb)),aa_num_nat(F,Y3)) ) ).

% in_measure
tff(fact_7685_in__measure,axiom,
    ! [Xb: nat,Y3: nat,F: fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y3),measure_nat(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Xb)),aa_nat_nat(F,Y3)) ) ).

% in_measure
tff(fact_7686_in__measure,axiom,
    ! [Xb: int,Y3: int,F: fun_int_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y3),measure_int(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,Xb)),aa_int_nat(F,Y3)) ) ).

% in_measure
tff(fact_7687_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
    <=> ! [M5: int] :
        ? [N4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M5),aa_int_int(abs_abs_int,N4))
          & aa_set_int_o(member_int(N4),S3) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_7688_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,Aa: nat,B: nat,Acc2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,Aa),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Aa),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,Aa),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_7689_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y3 )
     => ( Y3 = $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_7690_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,Aa: nat,B: nat] : aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(Aa,B)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_pj(F),Aa),B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_7691_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,Aa: nat,B: nat] : groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(Aa,B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_pk(F),Aa,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_7692_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,Aa: nat,B: nat] : groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(Aa,B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_pl(F),Aa),B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_7693_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,Aa: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(Aa,B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_pm(F),Aa),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_7694_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,Aa: nat,B: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(Aa,B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_pn(F),Aa),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_7695_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N4)
          & aa_set_nat_o(member_nat(N4),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_7696_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M2)
         => ? [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N5)
              & aa_set_nat_o(member_nat(N5),S3) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S3) ) ).

% unbounded_k_infinite
tff(fact_7697_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
          & aa_set_nat_o(member_nat(N4),S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_7698_sum__gp,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_complex,
        $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb)))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb))) ) ).

% sum_gp
tff(fact_7699_sum__gp,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_rat,
        $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ) ).

% sum_gp
tff(fact_7700_sum__gp,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_real,
        $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb)))),aa_real_real(minus_minus_real(one_one_real),Xb))) ) ).

% sum_gp
tff(fact_7701_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_7702_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_7703_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_7704_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7705_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_7706_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y3: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y3) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y3)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y3)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y3)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( (Y3)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Y3)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_7707_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7708_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7709_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7710_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Mb) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7711_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_num_int(numeral_numeral_int,V) )
    <=> ( Mb = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_7712_abs__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_7713_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_7714_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_7715_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_7716_negative__zle,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zle
tff(fact_7717_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = zero_zero_complex )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7718_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = zero_zero_rat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7719_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = zero_zero_int )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7720_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = zero_zero_real )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7721_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = zero_zero_nat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7722_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Mb) = zero_z3403309356797280102nteger )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7723_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7724_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_7725_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_7726_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_7727_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_7728_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_7729_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_7730_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_7731_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_7732_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_7733_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_7734_of__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% of_nat_0
tff(fact_7735_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_7736_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% of_nat_numeral
tff(fact_7737_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_7738_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% of_nat_numeral
tff(fact_7739_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_7740_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_7741_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7742_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7743_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7744_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7745_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7746_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7747_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7748_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7749_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7750_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7751_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_7752_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_7753_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_7754_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_7755_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_add
tff(fact_7756_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_7757_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_7758_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_7759_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_7760_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_7761_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_7762_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7763_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_7764_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_7765_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_7766_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_7767_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_7768_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7769_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_7770_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_7771_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_7772_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_7773_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_7774_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_7775_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_7776_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_7777_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_7778_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mult
tff(fact_7779_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,Mb)),Nb) ).

% of_nat_power
tff(fact_7780_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Nb) ).

% of_nat_power
tff(fact_7781_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),Nb) ).

% of_nat_power
tff(fact_7782_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),Nb) ).

% of_nat_power
tff(fact_7783_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),Nb) ).

% of_nat_power
tff(fact_7784_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,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_7785_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_7786_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_7787_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_7788_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_7789_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,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_7790_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_7791_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_7792_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_7793_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_7794_negative__zless,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zless
tff(fact_7795_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n3304061248610475627l_real((P)) ).

% of_nat_of_bool
tff(fact_7796_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_nat(zero_n2687167440665602831ol_nat,(P)) ).

% of_nat_of_bool
tff(fact_7797_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_nat_of_bool
tff(fact_7798_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n356916108424825756nteger((P)) ).

% of_nat_of_bool
tff(fact_7799_of__nat__sum,axiom,
    ! [F: fun_int_nat,A: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A)) = groups4538972089207619220nt_int(aTP_Lamm_po(F),A) ).

% of_nat_sum
tff(fact_7800_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A)) = groups7754918857620584856omplex(aTP_Lamm_pp(F),A) ).

% of_nat_sum
tff(fact_7801_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups3539618377306564664at_int(aTP_Lamm_pq(F),A) ).

% of_nat_sum
tff(fact_7802_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups7501900531339628137nteger(aTP_Lamm_pr(F),A) ).

% of_nat_sum
tff(fact_7803_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ps(F)),A) ).

% of_nat_sum
tff(fact_7804_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pt(F)),A) ).

% of_nat_sum
tff(fact_7805_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7806_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),zero_z3403309356797280102nteger)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7807_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7808_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7809_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7810_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Mb)) ).

% of_nat_Suc
tff(fact_7811_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb)) ).

% of_nat_Suc
tff(fact_7812_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% of_nat_Suc
tff(fact_7813_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(plus_plus_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).

% of_nat_Suc
tff(fact_7814_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) ).

% of_nat_Suc
tff(fact_7815_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Mb)) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)) ).

% of_nat_Suc
tff(fact_7816_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)),aa_num_real(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_7817_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_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_7818_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_7819_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_7820_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_7821_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_7822_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_7823_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_7824_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y3) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Y3 = 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_7825_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y3) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) )
    <=> ( Y3 = 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_7826_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Y3 = 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_7827_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y3) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
    <=> ( Y3 = 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_7828_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y3 = 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_7829_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y3) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Nb) )
    <=> ( Y3 = 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_7830_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7831_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7832_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: 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,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7833_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7834_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: 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,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7835_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Nb) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7836_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_7837_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_7838_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_7839_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_7840_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_7841_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_7842_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_7843_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_7844_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_7845_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_7846_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_7847_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_7848_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_7849_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_7850_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_7851_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_7852_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_7853_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_7854_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_7855_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_7856_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_7857_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_7858_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_7859_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_7860_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_7861_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: 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,Mb)) = 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,Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_7862_or__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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,Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_7863_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: 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,Mb)) = 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,Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(7)
tff(fact_7864_or__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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,Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(3)
tff(fact_7865_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_7866_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_7867_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_7868_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(aa_num_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_7869_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_7870_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,aa_num_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_7871_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_7872_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_7873_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(aa_num_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_7874_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_7875_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,aa_num_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_7876_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_7877_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_7878_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7879_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_7880_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7881_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_7882_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_7883_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7884_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_7885_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7886_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_7887_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_7888_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y3: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Y3) = aa_rat_rat(times_times_rat(Y3),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).

% mult_of_nat_commute
tff(fact_7889_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% mult_of_nat_commute
tff(fact_7890_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y3: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),Y3) = aa_real_real(times_times_real(Y3),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).

% mult_of_nat_commute
tff(fact_7891_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y3),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).

% mult_of_nat_commute
tff(fact_7892_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Y3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y3),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)) ).

% mult_of_nat_commute
tff(fact_7893_reals__Archimedean2,axiom,
    ! [Xb: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_7894_reals__Archimedean2,axiom,
    ! [Xb: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_7895_real__arch__simple,axiom,
    ! [Xb: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_7896_real__arch__simple,axiom,
    ! [Xb: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_7897_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_7898_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_7899_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_7900_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_7901_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_7902_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_7903_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_7904_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_7905_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_7906_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_7907_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_7908_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_7909_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_7910_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_7911_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_7912_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_7913_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_7914_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_7915_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_7916_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_7917_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Nb)) != zero_z3403309356797280102nteger ).

% of_nat_neq_0
tff(fact_7918_div__mult2__eq_H,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% div_mult2_eq'
tff(fact_7919_div__mult2__eq_H,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% div_mult2_eq'
tff(fact_7920_div__mult2__eq_H,axiom,
    ! [Aa: code_integer,Mb: nat,Nb: nat] : divide6298287555418463151nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = divide6298287555418463151nteger(divide6298287555418463151nteger(Aa,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% div_mult2_eq'
tff(fact_7921_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7922_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7923_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7924_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7925_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7926_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7927_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7928_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7929_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7930_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7931_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_7932_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_7933_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_7934_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_7935_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_7936_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7937_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7938_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = divide6298287555418463151nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7939_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_7940_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_7941_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_7942_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_7943_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% int_cases
tff(fact_7944_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N))
     => ( ! [N: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_7945_nat__int__comparison_I2_J,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_7946_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% zle_int
tff(fact_7947_nat__int__comparison_I3_J,axiom,
    ! [Aa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_7948_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_7949_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% zero_le_imp_eq_int
tff(fact_7950_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% nonneg_int_cases
tff(fact_7951_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))),Z) ).

% zadd_int_left
tff(fact_7952_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% int_plus
tff(fact_7953_int__ops_I5_J,axiom,
    ! [Aa: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_7954_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mod
tff(fact_7955_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mod
tff(fact_7956_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mod
tff(fact_7957_int__ops_I7_J,axiom,
    ! [Aa: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_7958_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z)
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_7959_zdiv__int,axiom,
    ! [Aa: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_7960_of__nat__max,axiom,
    ! [Xb: nat,Y3: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y3)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,semiri4216267220026989637d_enat(Xb)),semiri4216267220026989637d_enat(Y3)) ).

% of_nat_max
tff(fact_7961_of__nat__max,axiom,
    ! [Xb: nat,Y3: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y3)) ).

% of_nat_max
tff(fact_7962_of__nat__max,axiom,
    ! [Xb: nat,Y3: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y3)) = aa_real_real(ord_max_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,Y3)) ).

% of_nat_max
tff(fact_7963_of__nat__max,axiom,
    ! [Xb: nat,Y3: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(semiri1316708129612266289at_nat,Y3)) ).

% of_nat_max
tff(fact_7964_of__nat__max,axiom,
    ! [Xb: nat,Y3: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y3)) = ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb),aa_nat_Code_integer(semiri4939895301339042750nteger,Y3)) ).

% of_nat_max
tff(fact_7965_of__nat__and__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb)) = bit_se3949692690581998587nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_and_eq
tff(fact_7966_of__nat__and__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_and_eq
tff(fact_7967_of__nat__and__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_and_eq
tff(fact_7968_of__nat__or__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb)) = bit_se1080825931792720795nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_or_eq
tff(fact_7969_of__nat__or__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_or_eq
tff(fact_7970_of__nat__or__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_or_eq
tff(fact_7971_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_7972_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_7973_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_7974_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(6)
tff(fact_7975_or__not__num__neg_Osimps_I3_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(3)
tff(fact_7976_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_7977_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(5)
tff(fact_7978_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(9)
tff(fact_7979_ex__less__of__nat__mult,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_7980_ex__less__of__nat__mult,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_7981_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_7982_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_7983_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_7984_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_7985_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = minus_8373710615458151222nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ) ).

% of_nat_diff
tff(fact_7986_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ! [Y2: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).

% reals_Archimedean3
tff(fact_7987_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N: nat] : Mb != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( Mb != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_7988_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(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))) ).

% real_of_nat_div4
tff(fact_7989_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_7990_int__ops_I4_J,axiom,
    ! [Aa: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Aa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),one_one_int) ).

% int_ops(4)
tff(fact_7991_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_7992_int__zle__neg,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)))
    <=> ( ( Nb = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7993_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(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_7994_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_7995_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nonpos_int_cases
tff(fact_7996_int__sum,axiom,
    ! [F: fun_int_nat,A: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A)) = groups4538972089207619220nt_int(aTP_Lamm_po(F),A) ).

% int_sum
tff(fact_7997_int__sum,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups3539618377306564664at_int(aTP_Lamm_pq(F),A) ).

% int_sum
tff(fact_7998_or__not__num__neg_Osimps_I2_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(2)
tff(fact_7999_mod__mult2__eq_H,axiom,
    ! [Aa: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% mod_mult2_eq'
tff(fact_8000_mod__mult2__eq_H,axiom,
    ! [Aa: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).

% mod_mult2_eq'
tff(fact_8001_mod__mult2__eq_H,axiom,
    ! [Aa: code_integer,Mb: nat,Nb: nat] : modulo364778990260209775nteger(Aa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),modulo364778990260209775nteger(divide6298287555418463151nteger(Aa,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),modulo364778990260209775nteger(Aa,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb))) ).

% mod_mult2_eq'
tff(fact_8002_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(8)
tff(fact_8003_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_8004_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_8005_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_8006_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_8007_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_8008_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_8009_nat__less__real__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ).

% nat_less_real_le
tff(fact_8010_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_8011_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_8012_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_8013_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_8014_negD,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
     => ? [N: nat] : Xb = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_8015_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D: nat] : aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_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(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_8016_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_8017_nat__approx__posE,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_8018_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(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_8019_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_8020_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_8021_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_8022_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_8023_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_8024_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(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M2)),Xb)),C) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_8025_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_8026_or__not__num__neg_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y3: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y3 )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y3 != one ) ) )
       => ( ( ( Xb = one )
           => ! [M2: num] :
                ( ( Xaa = aa_num_num(bit0,M2) )
               => ( Y3 != aa_num_num(bit1,M2) ) ) )
         => ( ( ( Xb = one )
             => ! [M2: num] :
                  ( ( Xaa = aa_num_num(bit1,M2) )
                 => ( Y3 != aa_num_num(bit1,M2) ) ) )
           => ( ( ? [N: num] : Xb = aa_num_num(bit0,N)
               => ( ( Xaa = one )
                 => ( Y3 != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( Xb = aa_num_num(bit0,N) )
                   => ! [M2: num] :
                        ( ( Xaa = aa_num_num(bit0,M2) )
                       => ( Y3 != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
               => ( ! [N: num] :
                      ( ( Xb = aa_num_num(bit0,N) )
                     => ! [M2: num] :
                          ( ( Xaa = aa_num_num(bit1,M2) )
                         => ( Y3 != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
                 => ( ( ? [N: num] : Xb = aa_num_num(bit1,N)
                     => ( ( Xaa = one )
                       => ( Y3 != one ) ) )
                   => ( ! [N: num] :
                          ( ( Xb = aa_num_num(bit1,N) )
                         => ! [M2: num] :
                              ( ( Xaa = aa_num_num(bit0,M2) )
                             => ( Y3 != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xb = aa_num_num(bit1,N) )
                           => ! [M2: num] :
                                ( ( Xaa = aa_num_num(bit1,M2) )
                               => ( Y3 != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_8027_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(minus_minus_real(aa_real_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_8028_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xb: nat,Y3: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y3)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),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,Y3))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_8029_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_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_8030_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(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_realpow
tff(fact_8031_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(plus_plus_real(aa_real_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(plus_plus_real(Xb),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_8032_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(plus_plus_real(one_one_real),aa_real_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(plus_plus_real(one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_8033_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ! [Uv2: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv2)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_8034_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_8035_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum
tff(fact_8036_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum
tff(fact_8037_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_8038_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)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_8039_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_8040_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum
tff(fact_8041_double__arith__series,axiom,
    ! [Aa: complex,D: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_pu(Aa),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Aa)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),D))) ).

% double_arith_series
tff(fact_8042_double__arith__series,axiom,
    ! [Aa: rat,D: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_pv(Aa),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Aa)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),D))) ).

% double_arith_series
tff(fact_8043_double__arith__series,axiom,
    ! [Aa: 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_pw(Aa),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))),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D))) ).

% double_arith_series
tff(fact_8044_double__arith__series,axiom,
    ! [Aa: 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_px(Aa),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),D))) ).

% double_arith_series
tff(fact_8045_double__arith__series,axiom,
    ! [Aa: 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_py(Aa),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))),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D))) ).

% double_arith_series
tff(fact_8046_double__arith__series,axiom,
    ! [Aa: real,D: real,Nb: nat] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pz(Aa),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Aa)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),D))) ).

% double_arith_series
tff(fact_8047_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(plus_plus_real(one_one_real),aa_real_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(plus_plus_real(one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_8048_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8049_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8050_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_8051_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)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8052_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_8053_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8054_arith__series,axiom,
    ! [Aa: int,D: int,Nb: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_pw(Aa),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))),Aa)),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_8055_arith__series,axiom,
    ! [Aa: code_integer,D: code_integer,Nb: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_px(Aa),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa),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_8056_arith__series,axiom,
    ! [Aa: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_py(Aa),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))),Aa)),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_8057_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_8058_gauss__sum,axiom,
    ! [Nb: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8059_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_8060_sum__gp__offset,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Mb)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb))))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb))) ).

% sum_gp_offset
tff(fact_8061_sum__gp__offset,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_offset
tff(fact_8062_sum__gp__offset,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_real_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(minus_minus_real(one_one_real),Xb))) ).

% sum_gp_offset
tff(fact_8063_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Nb) = $ite(Nb = zero_zero_nat,zero_zero_complex,produc1917071388513777916omplex(aTP_Lamm_qa,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_8064_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_qb,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_8065_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_qc,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_8066_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_qd,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_8067_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_qe),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_8068_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = $ite(Nb = zero_zero_nat,zero_z3403309356797280102nteger,produc1830744345554046123nteger(aTP_Lamm_qf,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_8069_height__double__log__univ__size,axiom,
    ! [U: real,Dega: nat,Ta: vEBT_VEBT] :
      ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Dega) )
     => ( aa_nat_o(vEBT_invar_vebt(Ta),Dega)
       => 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(plus_plus_real(one_one_real),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_8070_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_qg,Xb)) ) ).

% monoseq_arctan_series
tff(fact_8071_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_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(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_8072_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(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(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_8073_ln__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_qh,Xb)) ) ) ) ).

% ln_series
tff(fact_8074_log__one,axiom,
    ! [Aa: real] : aa_real_real(log(Aa),one_one_real) = zero_zero_real ).

% log_one
tff(fact_8075_zero__less__log__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(log(Aa),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_8076_log__less__zero__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(log(Aa),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_8077_one__less__log__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(log(Aa),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_8078_log__less__one__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(log(Aa),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Aa) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_8079_log__less__cancel__iff,axiom,
    ! [Aa: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(Aa),Xb)),aa_real_real(log(Aa),Y3))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_8080_log__eq__one,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(log(Aa),Aa) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_8081_zero__le__log__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(log(Aa),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_8082_log__le__zero__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(log(Aa),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_8083_one__le__log__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(log(Aa),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_8084_log__le__one__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(log(Aa),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Aa) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_8085_log__le__cancel__iff,axiom,
    ! [Aa: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(Aa),Xb)),aa_real_real(log(Aa),Y3))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_8086_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_nx(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_8087_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_nz(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_8088_log__pow__cancel,axiom,
    ! [Aa: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(log(Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_8089_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_8090_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,Aa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),Aa))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,Aa)) ).

% complex_mod_triangle_ineq2
tff(fact_8091_less__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: 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)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Mb)) ) ) ).

% less_log_of_power
tff(fact_8092_log__of__power__eq,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ) ).

% log_of_power_eq
tff(fact_8093_log__base__change,axiom,
    ! [Aa: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(log(B),Xb) = aa_real_real(divide_divide_real(aa_real_real(log(Aa),Xb)),aa_real_real(log(Aa),B)) ) ) ) ).

% log_base_change
tff(fact_8094_le__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Mb)) ) ) ).

% le_log_of_power
tff(fact_8095_log__base__pow,axiom,
    ! [Aa: real,Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_real(log(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Nb)),Xb) = aa_real_real(divide_divide_real(aa_real_real(log(Aa),Xb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log_base_pow
tff(fact_8096_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(log(B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Xb)) ) ) ).

% log_nat_power
tff(fact_8097_log2__of__power__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ).

% log2_of_power_eq
tff(fact_8098_log__of__power__less,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_8099_log__mult,axiom,
    ! [Aa: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != 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),Y3)
           => ( aa_real_real(log(Aa),aa_real_real(times_times_real(Xb),Y3)) = aa_real_real(plus_plus_real(aa_real_real(log(Aa),Xb)),aa_real_real(log(Aa),Y3)) ) ) ) ) ) ).

% log_mult
tff(fact_8100_log__divide,axiom,
    ! [Aa: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != 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),Y3)
           => ( aa_real_real(log(Aa),aa_real_real(divide_divide_real(Xb),Y3)) = aa_real_real(minus_minus_real(aa_real_real(log(Aa),Xb)),aa_real_real(log(Aa),Y3)) ) ) ) ) ) ).

% log_divide
tff(fact_8101_log__of__power__le,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_8102_log__eq__div__ln__mult__log,axiom,
    ! [Aa: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != 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(log(Aa),Xb) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,Aa))),aa_real_real(log(B),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_8103_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_8104_less__log2__of__power,axiom,
    ! [Nb: nat,Mb: 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)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% less_log2_of_power
tff(fact_8105_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% le_log2_of_power
tff(fact_8106_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_8107_pred__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_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(vEBT_T_p_r_e_d2(Ta),Xb))),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_8108_succ__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_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(vEBT_T_s_u_c_c2(Ta),Xb))),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_8109_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_8110_pred__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_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(vEBT_T_p_r_e_d(Ta),Xb))),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_8111_insert__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_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(vEBT_T_i_n_s_e_r_t(Ta),Xb))),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_8112_succ__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_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(vEBT_T_s_u_c_c(Ta),Xb))),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_8113_member__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_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(vEBT_T_m_e_m_b_e_r(Ta),Xb))),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_8114_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_qi,Xb)) ) ) ).

% arctan_series
tff(fact_8115_norm__divide__numeral,axiom,
    ! [Aa: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(Aa),aa_num_real(numeral_numeral_real,W))) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,Aa)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_8116_norm__divide__numeral,axiom,
    ! [Aa: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_8117_norm__mult__numeral1,axiom,
    ! [W: num,Aa: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),Aa)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,Aa)) ).

% norm_mult_numeral1
tff(fact_8118_norm__mult__numeral1,axiom,
    ! [W: num,Aa: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),Aa)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,Aa)) ).

% norm_mult_numeral1
tff(fact_8119_norm__mult__numeral2,axiom,
    ! [Aa: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Aa),aa_num_real(numeral_numeral_real,W))) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,Aa)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_8120_norm__mult__numeral2,axiom,
    ! [Aa: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),numera6690914467698888265omplex(W))) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_8121_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_8122_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_8123_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_8124_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_8125_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_8126_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_8127_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_8128_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_8129_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_8130_norm__mult,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Xb),Y3)) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y3)) ).

% norm_mult
tff(fact_8131_norm__mult,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y3)) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y3)) ).

% norm_mult
tff(fact_8132_norm__divide,axiom,
    ! [Aa: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(Aa),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,B)) ).

% norm_divide
tff(fact_8133_norm__divide,axiom,
    ! [Aa: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,B)) ).

% norm_divide
tff(fact_8134_sum__norm__le,axiom,
    ! [S3: set_real,F: fun_real_complex,G: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),S3)
         => 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,S3))),groups8097168146408367636l_real(G,S3)) ) ).

% sum_norm_le
tff(fact_8135_sum__norm__le,axiom,
    ! [S3: set_set_nat,F: fun_set_nat_complex,G: fun_set_nat_real] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(F,X2))),aa_set_nat_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups8255218700646806128omplex(F,S3))),groups5107569545109728110t_real(G,S3)) ) ).

% sum_norm_le
tff(fact_8136_sum__norm__le,axiom,
    ! [S3: set_int,F: fun_int_complex,G: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),S3)
         => 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,S3))),groups8778361861064173332t_real(G,S3)) ) ).

% sum_norm_le
tff(fact_8137_sum__norm__le,axiom,
    ! [S3: set_nat,F: fun_nat_complex,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),S3)
         => 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,aa_set_nat_complex(groups2073611262835488442omplex(F),S3))),aa_set_nat_real(groups6591440286371151544t_real(G),S3)) ) ).

% sum_norm_le
tff(fact_8138_sum__norm__le,axiom,
    ! [S3: set_complex,F: fun_complex_complex,G: fun_complex_real] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),S3)
         => 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,S3))),groups5808333547571424918x_real(G,S3)) ) ).

% sum_norm_le
tff(fact_8139_sum__norm__le,axiom,
    ! [S3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),S3)
         => 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,aa_set_nat_real(groups6591440286371151544t_real(F),S3))),aa_set_nat_real(groups6591440286371151544t_real(G),S3)) ) ).

% sum_norm_le
tff(fact_8140_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_8141_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_8142_norm__sum,axiom,
    ! [F: fun_nat_complex,A: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qj(F)),A)) ).

% norm_sum
tff(fact_8143_norm__sum,axiom,
    ! [F: fun_complex_complex,A: set_complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A))),groups5808333547571424918x_real(aTP_Lamm_qk(F),A)) ).

% norm_sum
tff(fact_8144_norm__sum,axiom,
    ! [F: fun_nat_real,A: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ql(F)),A)) ).

% norm_sum
tff(fact_8145_norm__uminus__minus,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),Y3)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Y3)) ).

% norm_uminus_minus
tff(fact_8146_norm__uminus__minus,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y3)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y3)) ).

% norm_uminus_minus
tff(fact_8147_nonzero__norm__divide,axiom,
    ! [B: real,Aa: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(Aa),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_8148_nonzero__norm__divide,axiom,
    ! [B: complex,Aa: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_8149_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_8150_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_8151_norm__mult__less,axiom,
    ! [Xb: real,R2: real,Y3: real,S: 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,Y3)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Xb),Y3))),aa_real_real(times_times_real(R2),S)) ) ) ).

% norm_mult_less
tff(fact_8152_norm__mult__less,axiom,
    ! [Xb: complex,R2: real,Y3: complex,S: 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,Y3)),S)
       => 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),Y3))),aa_real_real(times_times_real(R2),S)) ) ) ).

% norm_mult_less
tff(fact_8153_norm__mult__ineq,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Xb),Y3))),aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y3))) ).

% norm_mult_ineq
tff(fact_8154_norm__mult__ineq,axiom,
    ! [Xb: complex,Y3: 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),Y3))),aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y3))) ).

% norm_mult_ineq
tff(fact_8155_norm__add__less,axiom,
    ! [Xb: real,R2: real,Y3: real,S: 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,Y3)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Y3))),aa_real_real(plus_plus_real(R2),S)) ) ) ).

% norm_add_less
tff(fact_8156_norm__add__less,axiom,
    ! [Xb: complex,R2: real,Y3: complex,S: 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,Y3)),S)
       => 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),Y3))),aa_real_real(plus_plus_real(R2),S)) ) ) ).

% norm_add_less
tff(fact_8157_norm__triangle__lt,axiom,
    ! [Xb: real,Y3: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y3))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Y3))),E) ) ).

% norm_triangle_lt
tff(fact_8158_norm__triangle__lt,axiom,
    ! [Xb: complex,Y3: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y3))),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),Y3))),E) ) ).

% norm_triangle_lt
tff(fact_8159_norm__triangle__mono,axiom,
    ! [Aa: real,R2: real,B: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Aa)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Aa),B))),aa_real_real(plus_plus_real(R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_8160_norm__triangle__mono,axiom,
    ! [Aa: complex,R2: real,B: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Aa)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S)
       => 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,Aa),B))),aa_real_real(plus_plus_real(R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_8161_norm__triangle__ineq,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Y3))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y3))) ).

% norm_triangle_ineq
tff(fact_8162_norm__triangle__ineq,axiom,
    ! [Xb: complex,Y3: 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),Y3))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y3))) ).

% norm_triangle_ineq
tff(fact_8163_norm__triangle__le,axiom,
    ! [Xb: real,Y3: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y3))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Y3))),E) ) ).

% norm_triangle_le
tff(fact_8164_norm__triangle__le,axiom,
    ! [Xb: complex,Y3: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y3))),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),Y3))),E) ) ).

% norm_triangle_le
tff(fact_8165_norm__add__leD,axiom,
    ! [Aa: 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(plus_plus_real(Aa),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Aa)),C)) ) ).

% norm_add_leD
tff(fact_8166_norm__add__leD,axiom,
    ! [Aa: 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,Aa),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Aa)),C)) ) ).

% norm_add_leD
tff(fact_8167_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_8168_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_8169_norm__triangle__sub,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Y3)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xb),Y3)))) ).

% norm_triangle_sub
tff(fact_8170_norm__triangle__sub,axiom,
    ! [Xb: complex,Y3: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Y3)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y3)))) ).

% norm_triangle_sub
tff(fact_8171_norm__triangle__ineq4,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Aa),B))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,B))) ).

% norm_triangle_ineq4
tff(fact_8172_norm__triangle__ineq4,axiom,
    ! [Aa: 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,Aa),B))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,B))) ).

% norm_triangle_ineq4
tff(fact_8173_norm__diff__triangle__le,axiom,
    ! [Xb: real,Y3: 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(minus_minus_real(Xb),Y3))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Y3),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xb),Z))),aa_real_real(plus_plus_real(E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_8174_norm__diff__triangle__le,axiom,
    ! [Xb: complex,Y3: 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),Y3))),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,Y3),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(plus_plus_real(E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_8175_norm__triangle__le__diff,axiom,
    ! [Xb: real,Y3: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y3))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xb),Y3))),E) ) ).

% norm_triangle_le_diff
tff(fact_8176_norm__triangle__le__diff,axiom,
    ! [Xb: complex,Y3: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y3))),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),Y3))),E) ) ).

% norm_triangle_le_diff
tff(fact_8177_norm__diff__ineq,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Aa),B))) ).

% norm_diff_ineq
tff(fact_8178_norm__diff__ineq,axiom,
    ! [Aa: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Aa),B))) ).

% norm_diff_ineq
tff(fact_8179_norm__triangle__ineq2,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Aa),B))) ).

% norm_triangle_ineq2
tff(fact_8180_norm__triangle__ineq2,axiom,
    ! [Aa: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),B))) ).

% norm_triangle_ineq2
tff(fact_8181_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_8182_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_8183_norm__diff__triangle__ineq,axiom,
    ! [Aa: 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(minus_minus_real(aa_real_real(plus_plus_real(Aa),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Aa),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_8184_norm__diff__triangle__ineq,axiom,
    ! [Aa: 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,Aa),B)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,C),D)))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_8185_norm__triangle__ineq3,axiom,
    ! [Aa: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,B)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Aa),B))) ).

% norm_triangle_ineq3
tff(fact_8186_norm__triangle__ineq3,axiom,
    ! [Aa: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,B)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Aa),B))) ).

% norm_triangle_ineq3
tff(fact_8187_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_8188_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_8189_norm__power__diff,axiom,
    ! [Z: real,W: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,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(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Mb)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_8190_norm__power__diff,axiom,
    ! [Z: complex,W: complex,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,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),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),Mb)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z),W)))) ) ) ).

% norm_power_diff
tff(fact_8191_suminf__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => ( suminf_real(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_8192_suminf__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => ( suminf_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)) ) ) ).

% suminf_geometric
tff(fact_8193_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_qm) = zero_zero_complex ).

% suminf_zero
tff(fact_8194_suminf__zero,axiom,
    suminf_real(aTP_Lamm_id) = zero_zero_real ).

% suminf_zero
tff(fact_8195_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_cu) = zero_zero_nat ).

% suminf_zero
tff(fact_8196_suminf__zero,axiom,
    suminf_int(aTP_Lamm_qn) = zero_zero_int ).

% suminf_zero
tff(fact_8197_heigt__uplog__rel,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_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(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% heigt_uplog_rel
tff(fact_8198_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_complex(F,N) = zero_zero_complex ) )
       => ( suminf_complex(F) = aa_set_nat_complex(groups2073611262835488442omplex(F),N3) ) ) ) ).

% suminf_finite
tff(fact_8199_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N3) ) ) ) ).

% suminf_finite
tff(fact_8200_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N3) ) ) ) ).

% suminf_finite
tff(fact_8201_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N3) ) ) ) ).

% suminf_finite
tff(fact_8202_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(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb)))) ) ) ).

% log_ceil_idem
tff(fact_8203_of__int__ceiling__cancel,axiom,
    ! [Xb: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_ceiling_cancel
tff(fact_8204_of__int__ceiling__cancel,axiom,
    ! [Xb: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_ceiling_cancel
tff(fact_8205_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_8206_ceiling__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_8207_ceiling__one,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_8208_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_8209_ceiling__add__of__int,axiom,
    ! [Xb: rat,Z: int] : aa_rat_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,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Z) ).

% ceiling_add_of_int
tff(fact_8210_ceiling__add__of__int,axiom,
    ! [Xb: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_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_8211_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_8212_ceiling__le__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_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_8213_zero__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% zero_less_ceiling
tff(fact_8214_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_8215_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),aa_num_real(numeral_numeral_real,V)) ) ).

% ceiling_le_numeral
tff(fact_8216_ceiling__le__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_8217_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,aa_num_real(numeral_numeral_real,V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_8218_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)),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_8219_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_8220_ceiling__less__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_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_8221_one__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% one_le_ceiling
tff(fact_8222_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_8223_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_8224_ceiling__le__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_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_8225_one__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ).

% one_less_ceiling
tff(fact_8226_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_8227_ceiling__add__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(Xb),aa_num_real(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_8228_ceiling__add__numeral,axiom,
    ! [Xb: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xb),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_8229_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_8230_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_8231_ceiling__add__one,axiom,
    ! [Xb: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_8232_ceiling__add__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_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_8233_ceiling__diff__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(Xb),aa_num_real(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_8234_ceiling__diff__numeral,axiom,
    ! [Xb: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xb),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_8235_ceiling__diff__one,axiom,
    ! [Xb: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_8236_ceiling__diff__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_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_8237_ceiling__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_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_8238_ceiling__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(power_power_rat(aa_num_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_8239_ceiling__less__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_8240_ceiling__less__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_8241_zero__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb) ) ).

% zero_le_ceiling
tff(fact_8242_zero__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb) ) ).

% zero_le_ceiling
tff(fact_8243_ceiling__divide__eq__div__numeral,axiom,
    ! [Aa: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,Aa)),aa_num_real(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,Aa))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_8244_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(minus_minus_real(aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_8245_ceiling__less__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_8246_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(minus_minus_real(aa_num_real(numeral_numeral_real,V)),one_one_real)),Xb) ) ).

% numeral_le_ceiling
tff(fact_8247_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)),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xb) ) ).

% numeral_le_ceiling
tff(fact_8248_ceiling__le__neg__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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_8249_ceiling__le__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_8250_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_8251_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_8252_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [Aa: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,Aa)),aa_num_real(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,Aa)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_8253_ceiling__less__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_8254_ceiling__less__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_8255_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_8256_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_8257_ceiling__mono,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y3)),aa_real_int(archim7802044766580827645g_real,Xb)) ) ).

% ceiling_mono
tff(fact_8258_ceiling__mono,axiom,
    ! [Y3: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y3)),aa_rat_int(archim2889992004027027881ng_rat,Xb)) ) ).

% ceiling_mono
tff(fact_8259_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_8260_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,aa_rat_int(archim2889992004027027881ng_rat,Xb))) ).

% le_of_int_ceiling
tff(fact_8261_ceiling__less__cancel,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Y3))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3) ) ).

% ceiling_less_cancel
tff(fact_8262_ceiling__less__cancel,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y3))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ).

% ceiling_less_cancel
tff(fact_8263_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_8264_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_8265_ceiling__le__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_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_8266_ceiling__le,axiom,
    ! [Xb: real,Aa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Aa))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),Aa) ) ).

% ceiling_le
tff(fact_8267_ceiling__le,axiom,
    ! [Xb: rat,Aa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Aa))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Aa) ) ).

% ceiling_le
tff(fact_8268_less__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(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_8269_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_8270_ceiling__add__le,axiom,
    ! [Xb: rat,Y3: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xb),Y3))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Y3))) ).

% ceiling_add_le
tff(fact_8271_ceiling__add__le,axiom,
    ! [Xb: real,Y3: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(Xb),Y3))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y3))) ).

% ceiling_add_le
tff(fact_8272_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(plus_plus_real(R2),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_8273_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,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(plus_plus_rat(R2),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_8274_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_8275_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_8276_ceiling__divide__eq__div,axiom,
    ! [Aa: int,B: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Aa)),aa_int_real(ring_1_of_int_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)),B)) ).

% ceiling_divide_eq_div
tff(fact_8277_ceiling__divide__eq__div,axiom,
    ! [Aa: int,B: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,Aa)),aa_int_rat(ring_1_of_int_rat,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)),B)) ).

% ceiling_divide_eq_div
tff(fact_8278_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Ta))
    <=> ! [I5: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,I5)),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,I5)) )
         => aa_int_o(P,I5) ) ) ).

% ceiling_split
tff(fact_8279_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,Ta))
    <=> ! [I5: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I5)),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,I5)) )
         => aa_int_o(P,I5) ) ) ).

% ceiling_split
tff(fact_8280_ceiling__eq__iff,axiom,
    ! [Xb: real,Aa: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xb) = Aa )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Aa)),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)) ) ) ).

% ceiling_eq_iff
tff(fact_8281_ceiling__eq__iff,axiom,
    ! [Xb: rat,Aa: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = Aa )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Aa)),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,Aa)) ) ) ).

% ceiling_eq_iff
tff(fact_8282_ceiling__unique,axiom,
    ! [Z: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_8283_ceiling__unique,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_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))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = Z ) ) ) ).

% ceiling_unique
tff(fact_8284_ceiling__correct,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_8285_ceiling__correct,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(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,aa_rat_int(archim2889992004027027881ng_rat,Xb))) ) ).

% ceiling_correct
tff(fact_8286_mult__ceiling__le,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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(times_times_real(Aa),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,Aa)),aa_real_int(archim7802044766580827645g_real,B))) ) ) ).

% mult_ceiling_le
tff(fact_8287_mult__ceiling__le,axiom,
    ! [Aa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(Aa),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,Aa)),aa_rat_int(archim2889992004027027881ng_rat,B))) ) ) ).

% mult_ceiling_le
tff(fact_8288_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(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_8289_ceiling__less__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_8290_le__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xb) ) ).

% le_ceiling_iff
tff(fact_8291_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(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb) ) ).

% le_ceiling_iff
tff(fact_8292_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(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P2),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_8293_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,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_8294_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(times_times_real(aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P2),Q2)))),one_one_real)),Q2)),P2) ) ).

% ceiling_divide_lower
tff(fact_8295_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(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q2)))),one_one_rat)),Q2)),P2) ) ).

% ceiling_divide_lower
tff(fact_8296_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(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_8297_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))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_8298_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(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_8299_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(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_8300_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(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_8301_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_qo) ).

% pi_series
tff(fact_8302_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( 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_complex_complex(aa_com3342990543769683525omplex(times_times_complex,H),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qq(H),Z),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8303_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Nb: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Z),H)),Nb)),aa_nat_rat(power_power_rat(Z),Nb))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qs(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8304_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_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(times_times_real(H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qu(H),Z),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8305_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_qi,Xb)) ) ).

% summable_arctan_series
tff(fact_8306_or__not__num__neg_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y3: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y3 )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Y3 = 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) )
                 => ( ( Y3 = 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) )
                   => ( ( Y3 = 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))) ) ) )
             => ( ! [N: num] :
                    ( ( Xb = aa_num_num(bit0,N) )
                   => ( ( Xaa = one )
                     => ( ( Y3 = aa_num_num(bit0,one) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),one)) ) ) )
               => ( ! [N: num] :
                      ( ( Xb = aa_num_num(bit0,N) )
                     => ! [M2: num] :
                          ( ( Xaa = aa_num_num(bit0,M2) )
                         => ( ( Y3 = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit0,M2))) ) ) )
                 => ( ! [N: num] :
                        ( ( Xb = aa_num_num(bit0,N) )
                       => ! [M2: num] :
                            ( ( Xaa = aa_num_num(bit1,M2) )
                           => ( ( Y3 = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit1,M2))) ) ) )
                   => ( ! [N: num] :
                          ( ( Xb = aa_num_num(bit1,N) )
                         => ( ( Xaa = one )
                           => ( ( Y3 = one )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),one)) ) ) )
                     => ( ! [N: num] :
                            ( ( Xb = aa_num_num(bit1,N) )
                           => ! [M2: num] :
                                ( ( Xaa = aa_num_num(bit0,M2) )
                               => ( ( Y3 = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit0,M2))) ) ) )
                       => ~ ! [N: num] :
                              ( ( Xb = aa_num_num(bit1,N) )
                             => ! [M2: num] :
                                  ( ( Xaa = aa_num_num(bit1,M2) )
                                 => ( ( Y3 = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit1,M2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_8307_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(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(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(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_8308_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I),set_or890127255671739683et_nat(K))
    <=> aa_set_nat_o(ord_less_set_nat(I),K) ) ).

% lessThan_iff
tff(fact_8309_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_8310_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( aa_set_num_o(member_num(I),set_ord_lessThan_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I),K) ) ).

% lessThan_iff
tff(fact_8311_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% lessThan_iff
tff(fact_8312_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( aa_set_int_o(member_int(I),aa_int_set_int(set_ord_lessThan_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) ) ).

% lessThan_iff
tff(fact_8313_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_8314_lessThan__iff,axiom,
    ! [I: $o,K: $o] :
      ( member_o((I),aa_o_set_o(set_ord_lessThan_o,(K)))
    <=> aa_o_o(ord_less_o((I)),(K)) ) ).

% lessThan_iff
tff(fact_8315_powr__one__eq__one,axiom,
    ! [Aa: real] : aa_real_real(powr_real(one_one_real),Aa) = one_one_real ).

% powr_one_eq_one
tff(fact_8316_summable__single,axiom,
    ! [I: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ir(I),F)) ).

% summable_single
tff(fact_8317_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qv(I),F)) ).

% summable_single
tff(fact_8318_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qw(I),F)) ).

% summable_single
tff(fact_8319_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_qx(I),F)) ).

% summable_single
tff(fact_8320_summable__zero,axiom,
    summable_complex(aTP_Lamm_qm) ).

% summable_zero
tff(fact_8321_summable__zero,axiom,
    summable_real(aTP_Lamm_id) ).

% summable_zero
tff(fact_8322_summable__zero,axiom,
    summable_nat(aTP_Lamm_cu) ).

% summable_zero
tff(fact_8323_summable__zero,axiom,
    summable_int(aTP_Lamm_qn) ).

% summable_zero
tff(fact_8324_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_ol(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_8325_summable__iff__shift,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qy(F),K))
    <=> summable_complex(F) ) ).

% summable_iff_shift
tff(fact_8326_lessThan__subset__iff,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_lessThan_rat(Xb)),set_ord_lessThan_rat(Y3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3) ) ).

% lessThan_subset_iff
tff(fact_8327_lessThan__subset__iff,axiom,
    ! [Xb: num,Y3: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_lessThan_num(Xb)),set_ord_lessThan_num(Y3))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y3) ) ).

% lessThan_subset_iff
tff(fact_8328_lessThan__subset__iff,axiom,
    ! [Xb: nat,Y3: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Y3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y3) ) ).

% lessThan_subset_iff
tff(fact_8329_lessThan__subset__iff,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_int_set_int(set_ord_lessThan_int,Xb)),aa_int_set_int(set_ord_lessThan_int,Y3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3) ) ).

% lessThan_subset_iff
tff(fact_8330_lessThan__subset__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or5984915006950818249n_real(Xb)),set_or5984915006950818249n_real(Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ).

% lessThan_subset_iff
tff(fact_8331_lessThan__subset__iff,axiom,
    ! [Xb: $o,Y3: $o] :
      ( ord_less_eq_set_o(aa_o_set_o(set_ord_lessThan_o,(Xb)),aa_o_set_o(set_ord_lessThan_o,(Y3)))
    <=> aa_o_o(ord_less_eq_o((Xb)),(Y3)) ) ).

% lessThan_subset_iff
tff(fact_8332_powr__zero__eq__one,axiom,
    ! [Xb: real] :
      aa_real_real(powr_real(Xb),zero_zero_real) = $ite(Xb = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_8333_powr__nonneg__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Aa),Xb)),zero_zero_real)
    <=> ( Aa = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_8334_powr__less__cancel__iff,axiom,
    ! [Xb: real,Aa: 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(powr_real(Xb),Aa)),aa_real_real(powr_real(Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B) ) ) ).

% powr_less_cancel_iff
tff(fact_8335_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qz(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_8336_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_kg(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_8337_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ra(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_8338_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_kp(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_8339_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rb(P),F)) ) ).

% summable_If_finite
tff(fact_8340_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_rc(P),F)) ) ).

% summable_If_finite
tff(fact_8341_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rd(P),F)) ) ).

% summable_If_finite
tff(fact_8342_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_re(P),F)) ) ).

% summable_If_finite
tff(fact_8343_summable__If__finite__set,axiom,
    ! [A: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rf(A),F)) ) ).

% summable_If_finite_set
tff(fact_8344_summable__If__finite__set,axiom,
    ! [A: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_rg(A),F)) ) ).

% summable_If_finite_set
tff(fact_8345_summable__If__finite__set,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rh(A),F)) ) ).

% summable_If_finite_set
tff(fact_8346_summable__If__finite__set,axiom,
    ! [A: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ri(A),F)) ) ).

% summable_If_finite_set
tff(fact_8347_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8348_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8349_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8350_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8351_powr__eq__one__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( ( aa_real_real(powr_real(Aa),Xb) = one_one_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_8352_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_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_8353_powr__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),one_one_real) = Xb ) ) ).

% powr_one
tff(fact_8354_powr__le__cancel__iff,axiom,
    ! [Xb: real,Aa: 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(powr_real(Xb),Aa)),aa_real_real(powr_real(Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B) ) ) ).

% powr_le_cancel_iff
tff(fact_8355_numeral__powr__numeral__real,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(powr_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_8356_log__powr__cancel,axiom,
    ! [Aa: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(log(Aa),aa_real_real(powr_real(Aa),Y3)) = Y3 ) ) ) ).

% log_powr_cancel
tff(fact_8357_powr__log__cancel,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(powr_real(Aa),aa_real_real(log(Aa),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_8358_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(power_power_real,C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_8359_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(power_power_complex,C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_8360_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(powr_real(Xb),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_8361_square__powr__half,axiom,
    ! [Xb: real] : aa_real_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% square_powr_half
tff(fact_8362_summable__norm__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ql(F))
     => summable_real(F) ) ).

% summable_norm_cancel
tff(fact_8363_summable__norm__cancel,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_qj(F))
     => summable_complex(F) ) ).

% summable_norm_cancel
tff(fact_8364_powr__powr,axiom,
    ! [Xb: real,Aa: real,B: real] : aa_real_real(powr_real(aa_real_real(powr_real(Xb),Aa)),B) = aa_real_real(powr_real(Xb),aa_real_real(times_times_real(Aa),B)) ).

% powr_powr
tff(fact_8365_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N3: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_8366_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N3: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_8367_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_8368_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_8369_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_rj,C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_8370_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_rk,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_8371_summable__mult,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_rl(F),C)) ) ).

% summable_mult
tff(fact_8372_summable__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_rm(F),C)) ) ).

% summable_mult
tff(fact_8373_summable__mult2,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_rn(F),C)) ) ).

% summable_mult2
tff(fact_8374_summable__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),C)) ) ).

% summable_mult2
tff(fact_8375_summable__add,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ro(F),G)) ) ) ).

% summable_add
tff(fact_8376_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_kk(F),G)) ) ) ).

% summable_add
tff(fact_8377_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kj(F),G)) ) ) ).

% summable_add
tff(fact_8378_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_rp(F),G)) ) ) ).

% summable_add
tff(fact_8379_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_oj(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_8380_summable__Suc__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_rq(F))
    <=> summable_complex(F) ) ).

% summable_Suc_iff
tff(fact_8381_summable__diff,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rr(F),G)) ) ) ).

% summable_diff
tff(fact_8382_summable__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_kn(F),G)) ) ) ).

% summable_diff
tff(fact_8383_summable__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ra(F),C)) ) ).

% summable_divide
tff(fact_8384_summable__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_kp(F),C)) ) ).

% summable_divide
tff(fact_8385_summable__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => summable_real(aTP_Lamm_ks(F)) ) ).

% summable_minus
tff(fact_8386_summable__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_complex(aTP_Lamm_rs(F)) ) ).

% summable_minus
tff(fact_8387_summable__minus__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ks(F))
    <=> summable_real(F) ) ).

% summable_minus_iff
tff(fact_8388_summable__minus__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_rs(F))
    <=> summable_complex(F) ) ).

% summable_minus_iff
tff(fact_8389_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_ol(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_8390_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(F)
     => summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qy(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_8391_summable__sum,axiom,
    ! [I6: set_complex,F: fun_co8378349014452122532t_real] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => summable_real(aa_com20334259897932645t_real(F,I2)) )
     => summable_real(aa_fun471735231769099483t_real(aTP_Lamm_ru(I6),F)) ) ).

% summable_sum
tff(fact_8392_summable__sum,axiom,
    ! [I6: set_real,F: fun_re4238438383274954274t_real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I6)
         => summable_real(aa_real_fun_nat_real(F,I2)) )
     => summable_real(aa_fun628238740774635481t_real(aTP_Lamm_rv(I6),F)) ) ).

% summable_sum
tff(fact_8393_summable__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => summable_real(aa_int_fun_nat_real(F,I2)) )
     => summable_real(aa_fun2842512154077127769t_real(aTP_Lamm_rx(I6),F)) ) ).

% summable_sum
tff(fact_8394_summable__sum,axiom,
    ! [I6: set_real,F: fun_re2084383175013312036omplex] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I6)
         => summable_complex(aa_rea1531280491672885477omplex(F,I2)) )
     => summable_complex(aa_fun7632768522510439005omplex(aTP_Lamm_rz(I6),F)) ) ).

% summable_sum
tff(fact_8395_summable__sum,axiom,
    ! [I6: set_nat,F: fun_na1142814060743658952omplex] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => summable_complex(aa_nat6454746962644518025omplex(F,I2)) )
     => summable_complex(aa_fun4886358607025424513omplex(aTP_Lamm_sb(I6),F)) ) ).

% summable_sum
tff(fact_8396_summable__sum,axiom,
    ! [I6: set_int,F: fun_in8056368165505483812omplex] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => summable_complex(aa_int4144929030551567077omplex(F,I2)) )
     => summable_complex(aa_fun3543561798013398749omplex(aTP_Lamm_sd(I6),F)) ) ).

% summable_sum
tff(fact_8397_summable__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_int] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => summable_int(aa_int_fun_nat_int(F,I2)) )
     => summable_int(aa_fun8710705519803144921at_int(aTP_Lamm_sf(I6),F)) ) ).

% summable_sum
tff(fact_8398_summable__sum,axiom,
    ! [I6: set_complex,F: fun_co3258106519541737638omplex] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => summable_complex(aa_com8181001469867462503omplex(F,I2)) )
     => summable_complex(aa_fun5003373910569599327omplex(aTP_Lamm_sh(I6),F)) ) ).

% summable_sum
tff(fact_8399_summable__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => summable_nat(aa_nat_fun_nat_nat(F,I2)) )
     => summable_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_si(I6),F)) ) ).

% summable_sum
tff(fact_8400_summable__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => summable_real(aa_nat_fun_nat_real(F,I2)) )
     => summable_real(aa_fun7039663905532832253t_real(aTP_Lamm_sj(I6),F)) ) ).

% summable_sum
tff(fact_8401_suminf__le__const,axiom,
    ! [F: fun_nat_int,Xb: int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_8402_suminf__le__const,axiom,
    ! [F: fun_nat_nat,Xb: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_8403_suminf__le__const,axiom,
    ! [F: fun_nat_real,Xb: real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_8404_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_if(F))
     => summable_real(F) ) ).

% summable_rabs_cancel
tff(fact_8405_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_sk(U)) ).

% lessThan_def
tff(fact_8406_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_sl,U)) ).

% lessThan_def
tff(fact_8407_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_sm,U)) ).

% lessThan_def
tff(fact_8408_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dj,U)) ).

% lessThan_def
tff(fact_8409_lessThan__def,axiom,
    ! [U: int] : aa_int_set_int(set_ord_lessThan_int,U) = collect_int(aa_int_fun_int_o(aTP_Lamm_sn,U)) ).

% lessThan_def
tff(fact_8410_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_so,U)) ).

% lessThan_def
tff(fact_8411_lessThan__def,axiom,
    ! [U: $o] : aa_o_set_o(set_ord_lessThan_o,(U)) = collect_o(aTP_Lamm_sp((U))) ).

% lessThan_def
tff(fact_8412_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,Xb: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8413_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,Xb: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8414_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,Xb: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8415_powser__insidea,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_sr(F),Z)) ) ) ).

% powser_insidea
tff(fact_8416_powser__insidea,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_st(F),Z)) ) ) ).

% powser_insidea
tff(fact_8417_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( summable_real(F)
       => ( summable_real(G)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),suminf_real(G)) ) ) ) ).

% suminf_le
tff(fact_8418_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G)) ) ) ) ).

% suminf_le
tff(fact_8419_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( summable_int(F)
       => ( summable_int(G)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),suminf_int(G)) ) ) ) ).

% suminf_le
tff(fact_8420_powr__ge__pzero,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(powr_real(Xb),Y3)) ).

% powr_ge_pzero
tff(fact_8421_powr__mono2,axiom,
    ! [Aa: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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),Y3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),Aa)),aa_real_real(powr_real(Y3),Aa)) ) ) ) ).

% powr_mono2
tff(fact_8422_powr__less__cancel,axiom,
    ! [Xb: real,Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xb),Aa)),aa_real_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,Aa),B) ) ) ).

% powr_less_cancel
tff(fact_8423_powr__less__mono,axiom,
    ! [Aa: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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(powr_real(Xb),Aa)),aa_real_real(powr_real(Xb),B)) ) ) ).

% powr_less_mono
tff(fact_8424_powr__mono,axiom,
    ! [Aa: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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(powr_real(Xb),Aa)),aa_real_real(powr_real(Xb),B)) ) ) ).

% powr_mono
tff(fact_8425_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(F)
     => ( suminf_complex(F) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qy(F),K))),aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_8426_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(plus_plus_real(suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ol(F),K))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_8427_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(F)
     => ( suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qy(F),K)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(F)),aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_8428_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ol(F),K)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_8429_lessThan__strict__subset__iff,axiom,
    ! [Mb: rat,Nb: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_ord_lessThan_rat(Mb)),set_ord_lessThan_rat(Nb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8430_lessThan__strict__subset__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_set_num_o(ord_less_set_num(set_ord_lessThan_num(Mb)),set_ord_lessThan_num(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8431_lessThan__strict__subset__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8432_lessThan__strict__subset__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_set_int_o(ord_less_set_int(aa_int_set_int(set_ord_lessThan_int,Mb)),aa_int_set_int(set_ord_lessThan_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8433_lessThan__strict__subset__iff,axiom,
    ! [Mb: real,Nb: real] :
      ( aa_set_real_o(ord_less_set_real(set_or5984915006950818249n_real(Mb)),set_or5984915006950818249n_real(Nb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_8434_lessThan__strict__subset__iff,axiom,
    ! [Mb: $o,Nb: $o] :
      ( ord_less_set_o(aa_o_set_o(set_ord_lessThan_o,(Mb)),aa_o_set_o(set_ord_lessThan_o,(Nb)))
    <=> aa_o_o(ord_less_o((Mb)),(Nb)) ) ).

% lessThan_strict_subset_iff
tff(fact_8435_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qz(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_8436_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_kg(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_8437_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_8438_summable__zero__power,axiom,
    summable_int(aa_int_fun_nat_int(power_power_int,zero_zero_int)) ).

% summable_zero_power
tff(fact_8439_summable__zero__power,axiom,
    summable_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex)) ).

% summable_zero_power
tff(fact_8440_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_8441_lessThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% lessThan_Suc
tff(fact_8442_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Nb: nat] :
      ( summable_int(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M2)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_8443_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( summable_nat(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M2)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_8444_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Nb: nat] :
      ( summable_real(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M2)) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_8445_suminf__mult,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_rl(F),C)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),suminf_complex(F)) ) ) ).

% suminf_mult
tff(fact_8446_suminf__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_rm(F),C)) = aa_real_real(times_times_real(C),suminf_real(F)) ) ) ).

% suminf_mult
tff(fact_8447_suminf__mult2,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(F)),C) = suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_rn(F),C)) ) ) ).

% suminf_mult2
tff(fact_8448_suminf__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( aa_real_real(times_times_real(suminf_real(F)),C) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),C)) ) ) ).

% suminf_mult2
tff(fact_8449_suminf__add,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,suminf_complex(F)),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ro(F),G)) ) ) ) ).

% suminf_add
tff(fact_8450_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(plus_plus_real(suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_kk(F),G)) ) ) ) ).

% suminf_add
tff(fact_8451_suminf__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kj(F),G)) ) ) ) ).

% suminf_add
tff(fact_8452_suminf__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_rp(F),G)) ) ) ) ).

% suminf_add
tff(fact_8453_suminf__diff,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(F)),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rr(F),G)) ) ) ) ).

% suminf_diff
tff(fact_8454_suminf__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(minus_minus_real(suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_kn(F),G)) ) ) ) ).

% suminf_diff
tff(fact_8455_suminf__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ra(F),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,suminf_complex(F)),C) ) ) ).

% suminf_divide
tff(fact_8456_suminf__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_kp(F),C)) = aa_real_real(divide_divide_real(suminf_real(F)),C) ) ) ).

% suminf_divide
tff(fact_8457_suminf__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_ks(F)) = aa_real_real(uminus_uminus_real,suminf_real(F)) ) ) ).

% suminf_minus
tff(fact_8458_suminf__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => ( suminf_complex(aTP_Lamm_rs(F)) = aa_complex_complex(uminus1482373934393186551omplex,suminf_complex(F)) ) ) ).

% suminf_minus
tff(fact_8459_suminf__sum,axiom,
    ! [I6: set_complex,F: fun_co8378349014452122532t_real] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => summable_real(aa_com20334259897932645t_real(F,I2)) )
     => ( suminf_real(aa_fun471735231769099483t_real(aTP_Lamm_ru(I6),F)) = groups5808333547571424918x_real(aTP_Lamm_su(F),I6) ) ) ).

% suminf_sum
tff(fact_8460_suminf__sum,axiom,
    ! [I6: set_real,F: fun_re4238438383274954274t_real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I6)
         => summable_real(aa_real_fun_nat_real(F,I2)) )
     => ( suminf_real(aa_fun628238740774635481t_real(aTP_Lamm_rv(I6),F)) = groups8097168146408367636l_real(aTP_Lamm_sv(F),I6) ) ) ).

% suminf_sum
tff(fact_8461_suminf__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => summable_real(aa_int_fun_nat_real(F,I2)) )
     => ( suminf_real(aa_fun2842512154077127769t_real(aTP_Lamm_rx(I6),F)) = groups8778361861064173332t_real(aTP_Lamm_sw(F),I6) ) ) ).

% suminf_sum
tff(fact_8462_suminf__sum,axiom,
    ! [I6: set_real,F: fun_re2084383175013312036omplex] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I6)
         => summable_complex(aa_rea1531280491672885477omplex(F,I2)) )
     => ( suminf_complex(aa_fun7632768522510439005omplex(aTP_Lamm_rz(I6),F)) = groups5754745047067104278omplex(aTP_Lamm_sx(F),I6) ) ) ).

% suminf_sum
tff(fact_8463_suminf__sum,axiom,
    ! [I6: set_nat,F: fun_na1142814060743658952omplex] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => summable_complex(aa_nat6454746962644518025omplex(F,I2)) )
     => ( suminf_complex(aa_fun4886358607025424513omplex(aTP_Lamm_sb(I6),F)) = aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_sy(F)),I6) ) ) ).

% suminf_sum
tff(fact_8464_suminf__sum,axiom,
    ! [I6: set_int,F: fun_in8056368165505483812omplex] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => summable_complex(aa_int4144929030551567077omplex(F,I2)) )
     => ( suminf_complex(aa_fun3543561798013398749omplex(aTP_Lamm_sd(I6),F)) = groups3049146728041665814omplex(aTP_Lamm_sz(F),I6) ) ) ).

% suminf_sum
tff(fact_8465_suminf__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_int] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I6)
         => summable_int(aa_int_fun_nat_int(F,I2)) )
     => ( suminf_int(aa_fun8710705519803144921at_int(aTP_Lamm_sf(I6),F)) = groups4538972089207619220nt_int(aTP_Lamm_ta(F),I6) ) ) ).

% suminf_sum
tff(fact_8466_suminf__sum,axiom,
    ! [I6: set_complex,F: fun_co3258106519541737638omplex] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I6)
         => summable_complex(aa_com8181001469867462503omplex(F,I2)) )
     => ( suminf_complex(aa_fun5003373910569599327omplex(aTP_Lamm_sh(I6),F)) = groups7754918857620584856omplex(aTP_Lamm_tb(F),I6) ) ) ).

% suminf_sum
tff(fact_8467_suminf__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => summable_nat(aa_nat_fun_nat_nat(F,I2)) )
     => ( suminf_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_si(I6),F)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_tc(F)),I6) ) ) ).

% suminf_sum
tff(fact_8468_suminf__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => summable_real(aa_nat_fun_nat_real(F,I2)) )
     => ( suminf_real(aa_fun7039663905532832253t_real(aTP_Lamm_sj(I6),F)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_td(F)),I6) ) ) ).

% suminf_sum
tff(fact_8469_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [K3: nat] : aa_set_nat_o(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_bounded
tff(fact_8470_finite__nat__iff__bounded,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
    <=> ? [K2: nat] : aa_set_nat_o(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_lessThan_nat,K2)) ) ).

% finite_nat_iff_bounded
tff(fact_8471_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Nb: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M2)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8472_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Nb: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M2)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8473_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Nb: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M2)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8474_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_nonneg
tff(fact_8475_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_nonneg
tff(fact_8476_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_nonneg
tff(fact_8477_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N4: nat] : aa_nat_real(F,N4) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8478_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N4: nat] : aa_nat_nat(F,N4) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8479_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N4: nat] : aa_nat_int(F,N4) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8480_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,N))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_pos
tff(fact_8481_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_pos
tff(fact_8482_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_pos
tff(fact_8483_powr__mono2_H,axiom,
    ! [Aa: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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),Y3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Y3),Aa)),aa_real_real(powr_real(Xb),Aa)) ) ) ) ).

% powr_mono2'
tff(fact_8484_powr__less__mono2,axiom,
    ! [Aa: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( 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),Y3)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xb),Aa)),aa_real_real(powr_real(Y3),Aa)) ) ) ) ).

% powr_less_mono2
tff(fact_8485_powr__inj,axiom,
    ! [Aa: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( ( aa_real_real(powr_real(Aa),Xb) = aa_real_real(powr_real(Aa),Y3) )
        <=> ( Xb = Y3 ) ) ) ) ).

% powr_inj
tff(fact_8486_gr__one__powr,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(powr_real(Xb),Y3)) ) ) ).

% gr_one_powr
tff(fact_8487_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,Aa: 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)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(powr_real(Xb),Aa)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_8488_powr__mono__both,axiom,
    ! [Aa: real,B: real,Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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),Y3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),Aa)),aa_real_real(powr_real(Y3),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_8489_powr__le1,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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(powr_real(Xb),Aa)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_8490_powr__divide,axiom,
    ! [Xb: real,Y3: real,Aa: 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),Y3)
       => ( aa_real_real(powr_real(aa_real_real(divide_divide_real(Xb),Y3)),Aa) = aa_real_real(divide_divide_real(aa_real_real(powr_real(Xb),Aa)),aa_real_real(powr_real(Y3),Aa)) ) ) ) ).

% powr_divide
tff(fact_8491_powr__mult,axiom,
    ! [Xb: real,Y3: real,Aa: 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),Y3)
       => ( aa_real_real(powr_real(aa_real_real(times_times_real(Xb),Y3)),Aa) = aa_real_real(times_times_real(aa_real_real(powr_real(Xb),Aa)),aa_real_real(powr_real(Y3),Aa)) ) ) ) ).

% powr_mult
tff(fact_8492_divide__powr__uminus,axiom,
    ! [Aa: real,B: real,C: real] : aa_real_real(divide_divide_real(Aa),aa_real_real(powr_real(B),C)) = aa_real_real(times_times_real(Aa),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_8493_log__powr,axiom,
    ! [Xb: real,B: real,Y3: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(log(B),aa_real_real(powr_real(Xb),Y3)) = aa_real_real(times_times_real(Y3),aa_real_real(log(B),Xb)) ) ) ).

% log_powr
tff(fact_8494_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_nx(F)) ).

% summable_0_powser
tff(fact_8495_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_nz(F)) ).

% summable_0_powser
tff(fact_8496_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_nx(F)) ).

% summable_zero_power'
tff(fact_8497_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_nz(F)) ).

% summable_zero_power'
tff(fact_8498_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_te(F)) ).

% summable_zero_power'
tff(fact_8499_ln__powr,axiom,
    ! [Xb: real,Y3: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(powr_real(Xb),Y3)) = aa_real_real(times_times_real(Y3),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_powr
tff(fact_8500_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tf(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8501_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_tg(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8502_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tf(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8503_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_tg(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8504_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,Mb: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_th(F),Mb),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8505_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,Mb: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ti(F),Mb),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8506_powr__add,axiom,
    ! [Xb: real,Aa: real,B: real] : aa_real_real(powr_real(Xb),aa_real_real(plus_plus_real(Aa),B)) = aa_real_real(times_times_real(aa_real_real(powr_real(Xb),Aa)),aa_real_real(powr_real(Xb),B)) ).

% powr_add
tff(fact_8507_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] : aa_real_real(powr_real(W),aa_real_real(minus_minus_real(Z1),Z22)) = aa_real_real(divide_divide_real(aa_real_real(powr_real(W),Z1)),aa_real_real(powr_real(W),Z22)) ).

% powr_diff
tff(fact_8508_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_qj(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_8509_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_if(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8510_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_8511_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tj(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_8512_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tk(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_8513_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_if(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_if(F))) ) ).

% summable_rabs
tff(fact_8514_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,aa_int_set_int(set_ord_lessThan_int,Nb))),groups4541462559716669496nt_nat(Q,aa_int_set_int(set_ord_lessThan_int,Nb))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_og(Q),P),aa_int_set_int(set_ord_lessThan_int,Nb)) ) ) ).

% sum_diff_distrib
tff(fact_8515_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,Nb: real] :
      ( ! [X2: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X2)),aa_real_nat(P,X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(Nb))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Nb))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_oe(Q),P),set_or5984915006950818249n_real(Nb)) ) ) ).

% sum_diff_distrib
tff(fact_8516_sum__diff__distrib,axiom,
    ! [Q: fun_o_nat,P: fun_o_nat,Nb: $o] :
      ( ! [X2: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Q,(X2))),aa_o_nat(P,(X2)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(P,aa_o_set_o(set_ord_lessThan_o,(Nb)))),groups8507830703676809646_o_nat(Q,aa_o_set_o(set_ord_lessThan_o,(Nb)))) = groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_tl(Q),P),aa_o_set_o(set_ord_lessThan_o,(Nb))) ) ) ).

% sum_diff_distrib
tff(fact_8517_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),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oh(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ) ) ).

% sum_diff_distrib
tff(fact_8518_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).

% suminf_pos2
tff(fact_8519_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).

% suminf_pos2
tff(fact_8520_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).

% suminf_pos2
tff(fact_8521_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F))
        <=> ? [I5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I5)) ) ) ) ).

% suminf_pos_iff
tff(fact_8522_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F))
        <=> ? [I5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I5)) ) ) ) ).

% suminf_pos_iff
tff(fact_8523_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F))
        <=> ? [I5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I5)) ) ) ) ).

% suminf_pos_iff
tff(fact_8524_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(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_8525_less__log__iff,axiom,
    ! [B: real,Xb: real,Y3: 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,Y3),aa_real_real(log(B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),Y3)),Xb) ) ) ) ).

% less_log_iff
tff(fact_8526_log__less__iff,axiom,
    ! [B: real,Xb: real,Y3: 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(log(B),Xb)),Y3)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(powr_real(B),Y3)) ) ) ) ).

% log_less_iff
tff(fact_8527_less__powr__iff,axiom,
    ! [B: real,Xb: real,Y3: 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(powr_real(B),Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),Xb)),Y3) ) ) ) ).

% less_powr_iff
tff(fact_8528_powr__less__iff,axiom,
    ! [B: real,Xb: real,Y3: 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(powr_real(B),Y3)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(log(B),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_8529_powser__inside,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Z)) ) ) ).

% powser_inside
tff(fact_8530_powser__inside,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Z)) ) ) ).

% powser_inside
tff(fact_8531_summable__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,C)) ) ).

% summable_geometric
tff(fact_8532_summable__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => summable_complex(aa_com8181001469867462503omplex(power_power_complex,C)) ) ).

% summable_geometric
tff(fact_8533_complete__algebra__summable__geometric,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ).

% complete_algebra_summable_geometric
tff(fact_8534_complete__algebra__summable__geometric,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),one_one_real)
     => summable_complex(aa_com8181001469867462503omplex(power_power_complex,Xb)) ) ).

% complete_algebra_summable_geometric
tff(fact_8535_suminf__split__head,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => ( suminf_complex(aTP_Lamm_rq(F)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(F)),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8536_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_oj(F)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8537_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_8538_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_8539_pi__half__neq__two,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_8540_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_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))),D3)))),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))),D3)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_8541_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_os(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8542_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ot(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8543_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_oi(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8544_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oj(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8545_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_tm(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_8546_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_tn(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_8547_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_to(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_8548_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_ou(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8549_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_ov(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8550_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ow(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(minus_minus_real(aa_nat_real(F,Mb)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8551_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_rat,Nb: nat,R2: rat] : aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(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_tp(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_8552_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,aa_nat_set_nat(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_tq(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_8553_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_Code_integer,Nb: nat,R2: code_integer] : minus_8373710615458151222nteger(groups7501900531339628137nteger(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),R2)) = groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_tr(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_8554_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_real,Nb: nat,R2: real] : aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),R2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(F),R2)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_8555_summable__norm,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ql(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(F))),suminf_real(aTP_Lamm_ql(F))) ) ).

% summable_norm
tff(fact_8556_summable__norm,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_qj(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(F))),suminf_real(aTP_Lamm_qj(F))) ) ).

% summable_norm
tff(fact_8557_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_oi(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_8558_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oj(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_8559_powr__minus__divide,axiom,
    ! [Xb: real,Aa: real] : aa_real_real(powr_real(Xb),aa_real_real(uminus_uminus_real,Aa)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(powr_real(Xb),Aa)) ).

% powr_minus_divide
tff(fact_8560_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I6: set_nat] :
      ( summable_int(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I6))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,I6)),suminf_int(F)) ) ) ) ).

% sum_le_suminf
tff(fact_8561_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I6: set_nat] :
      ( summable_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I6))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I6)),suminf_nat(F)) ) ) ) ).

% sum_le_suminf
tff(fact_8562_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I6: set_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I6))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),I6)),suminf_real(F)) ) ) ) ).

% sum_le_suminf
tff(fact_8563_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(divide_divide_real(one_one_real),Xb) ) ) ).

% powr_neg_one
tff(fact_8564_powr__mult__base,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(times_times_real(Xb),aa_real_real(powr_real(Xb),Y3)) = aa_real_real(powr_real(Xb),aa_real_real(plus_plus_real(one_one_real),Y3)) ) ) ).

% powr_mult_base
tff(fact_8565_powr__le__iff,axiom,
    ! [B: real,Xb: real,Y3: 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(powr_real(B),Y3)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(log(B),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_8566_le__powr__iff,axiom,
    ! [B: real,Xb: real,Y3: 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(powr_real(B),Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),Xb)),Y3) ) ) ) ).

% le_powr_iff
tff(fact_8567_log__le__iff,axiom,
    ! [B: real,Xb: real,Y3: 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(log(B),Xb)),Y3)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(powr_real(B),Y3)) ) ) ) ).

% log_le_iff
tff(fact_8568_le__log__iff,axiom,
    ! [B: real,Xb: real,Y3: 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,Y3),aa_real_real(log(B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),Y3)),Xb) ) ) ) ).

% le_log_iff
tff(fact_8569_pi__half__neq__zero,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_8570_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_8571_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_8572_power__diff__1__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),one_one_complex) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),one_one_complex)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_8573_power__diff__1__eq,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_8574_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),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_8575_power__diff__1__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_8576_one__diff__power__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_8577_one__diff__power__eq,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_8578_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),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_8579_one__diff__power__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_8580_geometric__sum,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( Xb != one_one_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),one_one_complex)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_8581_geometric__sum,axiom,
    ! [Xb: rat,Nb: nat] :
      ( ( Xb != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8582_geometric__sum,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real)),aa_real_real(minus_minus_real(Xb),one_one_real)) ) ) ).

% geometric_sum
tff(fact_8583_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_tf(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8584_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Z)) = aa_real_real(plus_plus_real(aa_nat_real(F,zero_zero_nat)),aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_tg(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8585_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Z))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_tf(F),Z))),Z) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8586_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Z))
     => ( aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_tg(F),Z))),Z) = aa_real_real(minus_minus_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8587_ln__powr__bound,axiom,
    ! [Xb: real,Aa: 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),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(divide_divide_real(aa_real_real(powr_real(Xb),Aa)),Aa)) ) ) ).

% ln_powr_bound
tff(fact_8588_ln__powr__bound2,axiom,
    ! [Xb: real,Aa: 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)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(aa_real_real(ln_ln_real,Xb)),Aa)),aa_real_real(times_times_real(aa_real_real(powr_real(Aa),Aa)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_8589_log__add__eq__powr,axiom,
    ! [B: real,Xb: real,Y3: 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(plus_plus_real(aa_real_real(log(B),Xb)),Y3) = aa_real_real(log(B),aa_real_real(times_times_real(Xb),aa_real_real(powr_real(B),Y3))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_8590_add__log__eq__powr,axiom,
    ! [B: real,Xb: real,Y3: 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(plus_plus_real(Y3),aa_real_real(log(B),Xb)) = aa_real_real(log(B),aa_real_real(times_times_real(aa_real_real(powr_real(B),Y3)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_8591_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_real(F)
       => ? [N7: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ol(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8592_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_complex(F)
       => ? [N7: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qy(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8593_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E: real] :
      ( summable_complex(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ~ ! [N7: nat] :
              ~ ! [M: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M)
                 => ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(M,N5)))),E) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8594_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E: real] :
      ( summable_real(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ~ ! [N7: nat] :
              ~ ! [M: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M)
                 => ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(M,N5)))),E) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8595_minus__log__eq__powr,axiom,
    ! [B: real,Xb: real,Y3: 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(minus_minus_real(Y3),aa_real_real(log(B),Xb)) = aa_real_real(log(B),aa_real_real(divide_divide_real(aa_real_real(powr_real(B),Y3)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_8596_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_sq(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8597_Abel__lemma,axiom,
    ! [R2: real,R0: real,Aa: fun_nat_complex,M6: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,R2),R0)
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Aa,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N))),M6)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_tt(R2),Aa)) ) ) ) ).

% Abel_lemma
tff(fact_8598_sum__gp__strict,axiom,
    ! [Xb: complex,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb))) ).

% sum_gp_strict
tff(fact_8599_sum__gp__strict,axiom,
    ! [Xb: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_strict
tff(fact_8600_sum__gp__strict,axiom,
    ! [Xb: real,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Nb),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_real_real(minus_minus_real(one_one_real),Xb))) ).

% sum_gp_strict
tff(fact_8601_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_8602_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tu(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tv(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_8603_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,Mb: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tw(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tx(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_8604_lemma__termdiff1,axiom,
    ! [Z: int,H: int,Mb: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ty(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_tz(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_8605_lemma__termdiff1,axiom,
    ! [Z: real,H: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ua(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ub(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_8606_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_8607_power__diff__sumr2,axiom,
    ! [Xb: complex,Nb: nat,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y3)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_uc(Xb),Nb),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_8608_power__diff__sumr2,axiom,
    ! [Xb: rat,Nb: nat,Y3: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Y3),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y3)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ud(Xb),Nb),Y3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_8609_power__diff__sumr2,axiom,
    ! [Xb: int,Nb: nat,Y3: 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,Y3),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y3)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ue(Xb),Nb),Y3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_8610_power__diff__sumr2,axiom,
    ! [Xb: real,Nb: nat,Y3: real] : aa_real_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,Y3),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),Y3)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_uf(Xb),Nb),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_8611_diff__power__eq__sum,axiom,
    ! [Xb: complex,Nb: nat,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),aa_nat_nat(suc,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y3)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ug(Xb),Nb),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8612_diff__power__eq__sum,axiom,
    ! [Xb: rat,Nb: nat,Y3: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(Y3),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y3)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_uh(Xb),Nb),Y3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8613_diff__power__eq__sum,axiom,
    ! [Xb: int,Nb: nat,Y3: 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,Y3),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),Y3)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ui(Xb),Nb),Y3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8614_diff__power__eq__sum,axiom,
    ! [Xb: real,Nb: nat,Y3: real] : aa_real_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,Y3),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),Y3)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_uj(Xb),Nb),Y3)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8615_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_8616_summable__ratio__test,axiom,
    ! [C: real,N3: nat,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N)))),aa_real_real(times_times_real(C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_8617_summable__ratio__test,axiom,
    ! [C: real,N3: nat,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N)))),aa_real_real(times_times_real(C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8618_arctan__ubound,axiom,
    ! [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_8619_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_8620_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_Code_integer,K5: code_integer,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,P4)),K5) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),K5)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8621_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_rat,K5: rat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P4)),K5) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),K5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8622_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_int,K5: int,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P4)),K5) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K5)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8623_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_nat,K5: nat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P4)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),K5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8624_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_real,K5: real,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P4)),K5) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),K5)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8625_log__minus__eq__powr,axiom,
    ! [B: real,Xb: real,Y3: 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(minus_minus_real(aa_real_real(log(B),Xb)),Y3) = aa_real_real(log(B),aa_real_real(times_times_real(Xb),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,Y3)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_8626_one__diff__power__eq_H,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_uk(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_8627_one__diff__power__eq_H,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ul(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_8628_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_um(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_8629_one__diff__power__eq_H,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_un(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_8630_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_8631_arctan__bounded,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y3))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_8632_arctan__lbound,axiom,
    ! [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y3)) ).

% arctan_lbound
tff(fact_8633_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(powr_real(Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_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_8634_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_uo(F),G)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_up(F)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uq(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_8635_machin__Euler,axiom,
    aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_8636_machin,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_8637_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_oi(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8638_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oj(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8639_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_8640_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ur),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8641_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_8642_arcosh__def,axiom,
    ! [Xb: real] : aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(Xb),aa_real_real(powr_real(aa_real_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(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_8643_summable__complex__of__real,axiom,
    ! [F: fun_nat_real] :
      ( summable_complex(aTP_Lamm_us(F))
    <=> summable_real(F) ) ).

% summable_complex_of_real
tff(fact_8644_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_8645_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_8646_of__real__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_8647_of__real__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,Xb) = one_one_complex )
    <=> ( Xb = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_8648_of__real__1,axiom,
    aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).

% of_real_1
tff(fact_8649_of__real__1,axiom,
    aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).

% of_real_1
tff(fact_8650_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% of_real_numeral
tff(fact_8651_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_num_real(numeral_numeral_real,W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_8652_of__real__mult,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(times_times_real(Xb),Y3)) = aa_real_real(times_times_real(aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y3)) ).

% of_real_mult
tff(fact_8653_of__real__mult,axiom,
    ! [Xb: real,Y3: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(Xb),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y3)) ).

% of_real_mult
tff(fact_8654_of__real__divide,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(Xb),Y3)) = aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y3)) ).

% of_real_divide
tff(fact_8655_of__real__divide,axiom,
    ! [Xb: real,Y3: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(divide_divide_real(Xb),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y3)) ).

% of_real_divide
tff(fact_8656_of__real__add,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(plus_plus_real(Xb),Y3)) = aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y3)) ).

% of_real_add
tff(fact_8657_of__real__add,axiom,
    ! [Xb: real,Y3: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(plus_plus_real(Xb),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y3)) ).

% of_real_add
tff(fact_8658_of__real__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(real_V1803761363581548252l_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_V1803761363581548252l_real,Xb)),Nb) ).

% of_real_power
tff(fact_8659_of__real__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(real_V4546457046886955230omplex,Xb)),Nb) ).

% of_real_power
tff(fact_8660_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8661_of__real__sum,axiom,
    ! [F: fun_complex_real,S: set_complex] : aa_real_complex(real_V4546457046886955230omplex,groups5808333547571424918x_real(F,S)) = groups7754918857620584856omplex(aTP_Lamm_ut(F),S) ).

% of_real_sum
tff(fact_8662_of__real__sum,axiom,
    ! [F: fun_nat_real,S: set_nat] : aa_real_complex(real_V4546457046886955230omplex,aa_set_nat_real(groups6591440286371151544t_real(F),S)) = aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_us(F)),S) ).

% of_real_sum
tff(fact_8663_of__real__sum,axiom,
    ! [F: fun_nat_real,S: set_nat] : aa_real_real(real_V1803761363581548252l_real,aa_set_nat_real(groups6591440286371151544t_real(F),S)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uu(F)),S) ).

% of_real_sum
tff(fact_8664_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_8665_sin__cos__squared__add3,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Xb))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Xb))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_8666_sin__cos__squared__add3,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Xb))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Xb))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_8667_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) ).

% of_real_neg_numeral
tff(fact_8668_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_8669_cos__of__real__pi,axiom,
    aa_real_real(cos_real,aa_real_real(real_V1803761363581548252l_real,pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_of_real_pi
tff(fact_8670_cos__of__real__pi,axiom,
    aa_complex_complex(cos_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cos_of_real_pi
tff(fact_8671_sin__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_8672_sin__npi2,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).

% sin_npi2
tff(fact_8673_sin__npi__int,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_npi_int
tff(fact_8674_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_8675_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_8676_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_8677_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_8678_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,Xb)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_8679_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,Xb)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_8680_norm__of__real__addn,axiom,
    ! [Xb: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,Xb)),aa_num_real(numeral_numeral_real,B))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(Xb),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_8681_norm__of__real__addn,axiom,
    ! [Xb: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,Xb)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(Xb),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_8682_cos__periodic,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xb) ).

% cos_periodic
tff(fact_8683_sin__periodic,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xb) ).

% sin_periodic
tff(fact_8684_cos__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(cos_real,Xb) ).

% cos_2pi_minus
tff(fact_8685_cos__npi2,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_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_8686_cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_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_8687_sin__cos__squared__add,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_8688_sin__cos__squared__add,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_8689_sin__cos__squared__add2,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_8690_sin__cos__squared__add2,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_8691_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_8692_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_8693_sin__of__real__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_of_real_pi_half
tff(fact_8694_sin__of__real__pi__half,axiom,
    aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_8695_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_8696_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_8697_sin__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_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_8698_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8699_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_8700_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_8701_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_8702_cos__npi__int,axiom,
    ! [Nb: int] :
      aa_real_real(cos_real,aa_real_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_8703_cos__int__times__real,axiom,
    ! [Mb: int,Xb: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Mb)),aa_real_real(real_V1803761363581548252l_real,Xb))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(cos_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Mb)),Xb))) ).

% cos_int_times_real
tff(fact_8704_cos__int__times__real,axiom,
    ! [Mb: int,Xb: real] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_int_complex(ring_17405671764205052669omplex,Mb)),aa_real_complex(real_V4546457046886955230omplex,Xb))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Mb)),Xb))) ).

% cos_int_times_real
tff(fact_8705_sin__add,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),Y3)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y3))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y3))) ).

% sin_add
tff(fact_8706_polar__Ex,axiom,
    ! [Xb: real,Y3: real] :
    ? [R3: real,A2: real] :
      ( ( Xb = aa_real_real(times_times_real(R3),aa_real_real(cos_real,A2)) )
      & ( Y3 = aa_real_real(times_times_real(R3),aa_real_real(sin_real,A2)) ) ) ).

% polar_Ex
tff(fact_8707_sin__diff,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(Xb),Y3)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y3))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y3))) ).

% sin_diff
tff(fact_8708_cos__one__sin__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) = one_one_complex )
     => ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_8709_cos__one__sin__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
     => ( aa_real_real(sin_real,Xb) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_8710_sin__int__times__real,axiom,
    ! [Mb: int,Xb: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Mb)),aa_real_real(real_V1803761363581548252l_real,Xb))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(sin_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Mb)),Xb))) ).

% sin_int_times_real
tff(fact_8711_sin__int__times__real,axiom,
    ! [Mb: int,Xb: real] : aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_int_complex(ring_17405671764205052669omplex,Mb)),aa_real_complex(real_V4546457046886955230omplex,Xb))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Mb)),Xb))) ).

% sin_int_times_real
tff(fact_8712_cos__diff,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(Xb),Y3)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y3))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y3))) ).

% cos_diff
tff(fact_8713_cos__add,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Y3)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y3))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y3))) ).

% cos_add
tff(fact_8714_sin__zero__norm__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_8715_sin__zero__norm__cos__one,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_8716_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_8717_sin__cos__eq,axiom,
    ! [Xb: real] : aa_real_real(sin_real,Xb) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) ).

% sin_cos_eq
tff(fact_8718_sin__cos__eq,axiom,
    ! [Xb: complex] : aa_complex_complex(sin_complex,Xb) = aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Xb)) ).

% sin_cos_eq
tff(fact_8719_cos__sin__eq,axiom,
    ! [Xb: real] : aa_real_real(cos_real,Xb) = aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) ).

% cos_sin_eq
tff(fact_8720_cos__sin__eq,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,Xb) = aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Xb)) ).

% cos_sin_eq
tff(fact_8721_sin__double,axiom,
    ! [Xb: complex] : aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,Xb))),aa_complex_complex(cos_complex,Xb)) ).

% sin_double
tff(fact_8722_sin__double,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,Xb))),aa_real_real(cos_real,Xb)) ).

% sin_double
tff(fact_8723_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
      & ( aa_real_real(sin_real,Y) = aa_real_real(sin_real,Xb) )
      & ( aa_real_real(cos_real,Y) = aa_real_real(cos_real,Xb) ) ) ).

% sincos_principal_value
tff(fact_8724_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_8725_minus__sin__cos__eq,axiom,
    ! [Xb: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) = aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_8726_minus__sin__cos__eq,axiom,
    ! [Xb: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,Xb)) = aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_8727_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_8728_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_8729_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_8730_sin__cos__le1,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y3))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y3))))),one_one_real) ).

% sin_cos_le1
tff(fact_8731_summable__of__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => summable_real(aTP_Lamm_uu(X5)) ) ).

% summable_of_real
tff(fact_8732_summable__of__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => summable_complex(aTP_Lamm_us(X5)) ) ).

% summable_of_real
tff(fact_8733_cos__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_8734_cos__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_8735_sin__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_8736_sin__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_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_squared_eq
tff(fact_8737_nonzero__of__real__divide,axiom,
    ! [Y3: real,Xb: real] :
      ( ( Y3 != zero_zero_real )
     => ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(Xb),Y3)) = aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y3)) ) ) ).

% nonzero_of_real_divide
tff(fact_8738_nonzero__of__real__divide,axiom,
    ! [Y3: real,Xb: real] :
      ( ( Y3 != zero_zero_real )
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(divide_divide_real(Xb),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y3)) ) ) ).

% nonzero_of_real_divide
tff(fact_8739_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_8740_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_8741_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_8742_cos__inj__pi,axiom,
    ! [Xb: real,Y3: 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),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
           => ( ( aa_real_real(cos_real,Xb) = aa_real_real(cos_real,Y3) )
             => ( Xb = Y3 ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_8743_cos__mono__le__eq,axiom,
    ! [Xb: real,Y3: 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),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y3))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_8744_cos__monotone__0__pi__le,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),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,Y3)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_8745_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_8746_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_8747_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_8748_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_8749_cos__diff__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Z),W)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_8750_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_8751_sin__diff__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_8752_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_8753_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_8754_cos__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_8755_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_8756_sin__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_8757_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_8758_sin__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_8759_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_8760_cos__double,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_8761_cos__double,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_8762_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_8763_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_8764_suminf__of__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( aa_real_real(real_V1803761363581548252l_real,suminf_real(X5)) = suminf_real(aTP_Lamm_uu(X5)) ) ) ).

% suminf_of_real
tff(fact_8765_suminf__of__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( aa_real_complex(real_V4546457046886955230omplex,suminf_real(X5)) = suminf_complex(aTP_Lamm_us(X5)) ) ) ).

% suminf_of_real
tff(fact_8766_norm__less__p1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,Xb))),one_one_real))) ).

% norm_less_p1
tff(fact_8767_norm__less__p1,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Xb))),one_one_complex))) ).

% norm_less_p1
tff(fact_8768_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_8769_cos__mono__less__eq,axiom,
    ! [Xb: real,Y3: 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),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y3))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8770_cos__monotone__0__pi,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),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,Y3)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8771_cos__monotone__minus__pi__0_H,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),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,Y3)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_8772_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I5: int] : Xb = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I5)),pi) ) ).

% sin_zero_iff_int2
tff(fact_8773_sincos__total__pi,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
     => ( ( aa_real_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,Y3),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) )
            & ( Y3 = aa_real_real(sin_real,T3) ) ) ) ) ).

% sincos_total_pi
tff(fact_8774_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_8775_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_8776_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_8777_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_8778_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_8779_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X2) = zero_zero_real )
      & ! [Y2: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y2) = zero_zero_real ) )
         => ( Y2 = X2 ) ) ) ).

% cos_is_zero
tff(fact_8780_norm__of__real__diff,axiom,
    ! [B: real,Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(real_V1803761363581548252l_real,B)),aa_real_real(real_V1803761363581548252l_real,Aa)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),Aa))) ).

% norm_of_real_diff
tff(fact_8781_norm__of__real__diff,axiom,
    ! [B: real,Aa: 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_real_complex(real_V4546457046886955230omplex,B)),aa_real_complex(real_V4546457046886955230omplex,Aa)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),Aa))) ).

% norm_of_real_diff
tff(fact_8782_cos__monotone__minus__pi__0,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),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,Y3)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8783_cos__total,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),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) = Y3 )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
                  & ( aa_real_real(cos_real,Y2) = Y3 ) )
               => ( Y2 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_8784_sincos__total__pi__half,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( ( aa_real_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,Y3),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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xb = aa_real_real(cos_real,T3) )
              & ( Y3 = aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_8785_sincos__total__2pi__le,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_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,Y3),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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xb = aa_real_real(cos_real,T3) )
          & ( Y3 = aa_real_real(sin_real,T3) ) ) ) ).

% sincos_total_2pi_le
tff(fact_8786_sincos__total__2pi,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_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,Y3),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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xb = aa_real_real(cos_real,T3) )
               => ( Y3 != aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8787_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(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8788_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_8789_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_8790_cos__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_8791_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_8792_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_8793_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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_8794_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_8795_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_30
tff(fact_8796_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_8797_sin__inj__pi,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Y3) )
             => ( Xb = Y3 ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_8798_sin__mono__le__eq,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y3))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_8799_sin__monotone__2pi__le,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y3)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_8800_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_60
tff(fact_8801_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ? [X: int] : Xb = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_8802_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_8803_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_8804_cos__treble__cos,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit1,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,Xb))) ).

% cos_treble_cos
tff(fact_8805_cos__treble__cos,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),Xb)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,Xb))) ).

% cos_treble_cos
tff(fact_8806_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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_8807_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),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_8808_sin__monotone__2pi,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y3)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_8809_sin__mono__less__eq,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y3))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8810_sin__total,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),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(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X2) = Y3 )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y2) = Y3 ) )
               => ( Y2 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_8811_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_8812_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_8813_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ( ? [X: nat] : Xb = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X: nat] : Xb = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_8814_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(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8815_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I5: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I5)
          & ( Xb = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I5)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_8816_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ? [I5: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I5)
          & ( Xb = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I5)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_8817_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 )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_8818_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_8819_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 )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_8820_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_8821_arsinh__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(Xb),aa_real_real(powr_real(aa_real_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)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arsinh_def
tff(fact_8822_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(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_uv,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,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_8823_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(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_uv,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,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_8824_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(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_uv,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,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_8825_tan__double,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,Xb))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8826_tan__double,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,Xb))),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8827_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_8828_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_8829_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_8830_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_8831_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_8832_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_8833_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_8834_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_8835_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_8836_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_8837_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_8838_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_8839_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_8840_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_8841_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_8842_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_Suc
tff(fact_8843_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_Suc
tff(fact_8844_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% fact_Suc
tff(fact_8845_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_Suc
tff(fact_8846_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_Suc
tff(fact_8847_tan__npi,axiom,
    ! [Nb: nat] : aa_real_real(tan_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_8848_tan__periodic__n,axiom,
    ! [Xb: real,Nb: num] : aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_n
tff(fact_8849_tan__periodic__nat,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_nat
tff(fact_8850_tan__periodic__int,axiom,
    ! [Xb: real,I: int] : aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),aa_real_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_8851_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8852_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8853_fact__2,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8854_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8855_fact__2,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8856_tan__periodic,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic
tff(fact_8857_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_zero
tff(fact_8858_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_zero
tff(fact_8859_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_zero
tff(fact_8860_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_zero
tff(fact_8861_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_gt_zero
tff(fact_8862_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_gt_zero
tff(fact_8863_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_gt_zero
tff(fact_8864_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_gt_zero
tff(fact_8865_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),zero_zero_rat) ).

% fact_not_neg
tff(fact_8866_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),zero_zero_int) ).

% fact_not_neg
tff(fact_8867_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),zero_zero_real) ).

% fact_not_neg
tff(fact_8868_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),zero_zero_nat) ).

% fact_not_neg
tff(fact_8869_fact__ge__1,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_1
tff(fact_8870_fact__ge__1,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_1
tff(fact_8871_fact__ge__1,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_1
tff(fact_8872_fact__ge__1,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_1
tff(fact_8873_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% fact_mono
tff(fact_8874_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% fact_mono
tff(fact_8875_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% fact_mono
tff(fact_8876_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono
tff(fact_8877_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).

% fact_dvd
tff(fact_8878_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) ) ).

% fact_dvd
tff(fact_8879_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).

% fact_dvd
tff(fact_8880_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).

% fact_dvd
tff(fact_8881_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ) ).

% fact_less_mono
tff(fact_8882_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).

% fact_less_mono
tff(fact_8883_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).

% fact_less_mono
tff(fact_8884_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono
tff(fact_8885_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8886_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8887_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8888_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8889_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8890_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_8891_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,Nb),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_8892_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_8893_fact__le__power,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8894_fact__le__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8895_fact__le__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8896_fact__le__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8897_fact__le__power,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8898_tan__def,axiom,
    ! [X3: complex] : aa_complex_complex(tan_complex,X3) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(sin_complex,X3)),aa_complex_complex(cos_complex,X3)) ).

% tan_def
tff(fact_8899_tan__def,axiom,
    ! [X3: real] : aa_real_real(tan_real,X3) = aa_real_real(divide_divide_real(aa_real_real(sin_real,X3)),aa_real_real(cos_real,X3)) ).

% tan_def
tff(fact_8900_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ) ).

% choose_dvd
tff(fact_8901_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% choose_dvd
tff(fact_8902_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% choose_dvd
tff(fact_8903_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% choose_dvd
tff(fact_8904_choose__dvd,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(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% choose_dvd
tff(fact_8905_fact__numeral,axiom,
    ! [K: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(K)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8906_fact__numeral,axiom,
    ! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8907_fact__numeral,axiom,
    ! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8908_fact__numeral,axiom,
    ! [K: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8909_fact__numeral,axiom,
    ! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8910_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_8911_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_8912_fact__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8913_fact__code,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,Nb) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8914_fact__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8915_fact__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8916_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8917_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8918_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8919_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_Code_integer(semiri3624122377584611663nteger,Mb) = $ite(Mb = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8920_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_real(semiri2265585572941072030t_real,Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8921_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8922_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8923_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8924_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8925_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8926_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8927_lemma__tan__total,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
     => ? [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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(tan_real,X2)) ) ) ).

% lemma_tan_total
tff(fact_8928_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_gt_zero
tff(fact_8929_lemma__tan__total1,axiom,
    ! [Y3: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X2) = Y3 ) ) ).

% lemma_tan_total1
tff(fact_8930_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y3))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8931_tan__monotone_H,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y3)),aa_real_real(tan_real,Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8932_tan__monotone,axiom,
    ! [Y3: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y3)),aa_real_real(tan_real,Xb)) ) ) ) ).

% tan_monotone
tff(fact_8933_tan__total,axiom,
    ! [Y3: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X2) = Y3 )
      & ! [Y2: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y2) = Y3 ) )
         => ( Y2 = X2 ) ) ) ).

% tan_total
tff(fact_8934_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(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_8935_tan__inverse,axiom,
    ! [Y3: real] : aa_real_real(divide_divide_real(one_one_real),aa_real_real(tan_real,Y3)) = aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y3)) ).

% tan_inverse
tff(fact_8936_add__tan__eq,axiom,
    ! [Xb: complex,Y3: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y3) != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y3))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y3))) ) ) ) ).

% add_tan_eq
tff(fact_8937_add__tan__eq,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y3) != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y3)) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),Y3))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y3))) ) ) ) ).

% add_tan_eq
tff(fact_8938_tan__total__pos,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
     => ? [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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X2) = Y3 ) ) ) ).

% tan_total_pos
tff(fact_8939_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_8940_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),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_8941_tan__mono__le,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y3)) ) ) ) ).

% tan_mono_le
tff(fact_8942_tan__mono__le__eq,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y3))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8943_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(divide_divide_real(pi),aa_num_real(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_8944_arctan__unique,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xb) = Y3 )
         => ( aa_real_real(arctan,Y3) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_8945_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_8946_arctan,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y3))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y3)) = Y3 ) ) ).

% arctan
tff(fact_8947_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_uw(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_8948_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ux(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_8949_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_rat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_uy(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_8950_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_nat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_uz(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_8951_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_int_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_va(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_8952_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_vb(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(B7),aa_real_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_8953_tan__add,axiom,
    ! [Xb: complex,Y3: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y3) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y3)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y3))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y3)))) ) ) ) ) ).

% tan_add
tff(fact_8954_tan__add,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y3) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Y3)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),Y3)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y3))),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y3)))) ) ) ) ) ).

% tan_add
tff(fact_8955_tan__diff,axiom,
    ! [Xb: complex,Y3: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y3) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y3)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y3))),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y3)))) ) ) ) ) ).

% tan_diff
tff(fact_8956_tan__diff,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y3) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(minus_minus_real(Xb),Y3)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(minus_minus_real(Xb),Y3)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y3))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y3)))) ) ) ) ) ).

% tan_diff
tff(fact_8957_lemma__tan__add1,axiom,
    ! [Xb: complex,Y3: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y3) != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y3))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y3))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y3))) ) ) ) ).

% lemma_tan_add1
tff(fact_8958_lemma__tan__add1,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y3) != zero_zero_real )
       => ( aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y3))) = aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Y3))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y3))) ) ) ) ).

% lemma_tan_add1
tff(fact_8959_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(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z3) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_8960_tan__half,axiom,
    ! [Xb: complex] : aa_complex_complex(tan_complex,Xb) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),one_one_complex)) ).

% tan_half
tff(fact_8961_tan__half,axiom,
    ! [Xb: real] : aa_real_real(tan_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real)) ).

% tan_half
tff(fact_8962_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(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_8963_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(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vc,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,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_8964_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(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vc,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,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_8965_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(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vc,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,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_8966_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] : aa_real_real(sin_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vc,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,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_8967_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(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_8968_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_8969_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_8970_fact__less__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_8971_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_8972_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_8973_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_8974_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_8975_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_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_8976_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_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_8977_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(tan_real,Xb)),aa_real_real(sqrt,aa_real_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_8978_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xb) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_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_8979_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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T3)),aa_real_real(sin_real,T3)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8980_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(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vd,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_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_8981_real__sqrt__eq__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_real(sqrt,Xb) = aa_real_real(sqrt,Y3) )
    <=> ( Xb = Y3 ) ) ).

% real_sqrt_eq_iff
tff(fact_8982_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_8983_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_8984_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ).

% real_sqrt_less_iff
tff(fact_8985_real__sqrt__le__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ).

% real_sqrt_le_iff
tff(fact_8986_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_8987_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_8988_exp__le__cancel__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ).

% exp_le_cancel_iff
tff(fact_8989_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_8990_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_8991_real__sqrt__gt__0__iff,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ).

% real_sqrt_gt_0_iff
tff(fact_8992_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_8993_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_8994_real__sqrt__ge__0__iff,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3) ) ).

% real_sqrt_ge_0_iff
tff(fact_8995_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_8996_real__sqrt__gt__1__iff,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y3) ) ).

% real_sqrt_gt_1_iff
tff(fact_8997_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_8998_real__sqrt__ge__1__iff,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y3) ) ).

% real_sqrt_ge_1_iff
tff(fact_8999_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_9000_real__sqrt__abs2,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(times_times_real(Xb),Xb)) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs2
tff(fact_9001_real__sqrt__mult__self,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(aa_real_real(sqrt,Aa)),aa_real_real(sqrt,Aa)) = aa_real_real(abs_abs_real,Aa) ).

% real_sqrt_mult_self
tff(fact_9002_real__sqrt__four,axiom,
    aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_9003_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_9004_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_9005_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_9006_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_9007_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_9008_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_9009_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_9010_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_9011_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Y3: real,Xaa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_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(times_times_real(aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_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_9012_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_9013_exp__times__arg__commute,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(exp_complex,A)),A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(exp_complex,A)) ).

% exp_times_arg_commute
tff(fact_9014_exp__times__arg__commute,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(exp_real,A)),A) = aa_real_real(times_times_real(A),aa_real_real(exp_real,A)) ).

% exp_times_arg_commute
tff(fact_9015_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_9016_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y3)) ) ).

% real_sqrt_less_mono
tff(fact_9017_real__sqrt__mult,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(sqrt,aa_real_real(times_times_real(Xb),Y3)) = aa_real_real(times_times_real(aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y3)) ).

% real_sqrt_mult
tff(fact_9018_real__sqrt__divide,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(sqrt,aa_real_real(divide_divide_real(Xb),Y3)) = aa_real_real(divide_divide_real(aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y3)) ).

% real_sqrt_divide
tff(fact_9019_real__sqrt__le__mono,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y3)) ) ).

% real_sqrt_le_mono
tff(fact_9020_norm__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Xb))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,Xb))) ).

% norm_exp
tff(fact_9021_norm__exp,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Xb))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,Xb))) ).

% norm_exp
tff(fact_9022_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_9023_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_9024_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_9025_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_9026_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_9027_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_9028_Complex__eq__numeral,axiom,
    ! [Aa: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(Aa),B) = numera6690914467698888265omplex(W) )
    <=> ( ( Aa = aa_num_real(numeral_numeral_real,W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_9029_Complex__mult__complex__of__real,axiom,
    ! [Xb: real,Y3: real,R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(Xb),Y3)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(times_times_real(Xb),R2)),aa_real_real(times_times_real(Y3),R2)) ).

% Complex_mult_complex_of_real
tff(fact_9030_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xb: real,Y3: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Y3)) = aa_real_complex(complex2(aa_real_real(times_times_real(R2),Xb)),aa_real_real(times_times_real(R2),Y3)) ).

% complex_of_real_mult_Complex
tff(fact_9031_complex__norm,axiom,
    ! [Xb: real,Y3: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xb),Y3)) = aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_9032_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_9033_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(divide_divide_real(Xb),aa_real_real(sqrt,Xb)) = aa_real_real(sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_9034_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(Xb),Y3))),aa_real_real(plus_plus_real(aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y3))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_9035_exp__ge__add__one__self,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),Xb)),aa_real_real(exp_real,Xb)) ).

% exp_ge_add_one_self
tff(fact_9036_le__real__sqrt__sumsq,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y3),Y3)))) ).

% le_real_sqrt_sumsq
tff(fact_9037_Complex__eq__neg__numeral,axiom,
    ! [Aa: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(Aa),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( Aa = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_9038_log__ln,axiom,
    ! [Xb: real] : aa_real_real(ln_ln_real,Xb) = aa_real_real(log(aa_real_real(exp_real,one_one_real)),Xb) ).

% log_ln
tff(fact_9039_complex__mult,axiom,
    ! [Aa: real,B: real,C: real,D: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(Aa),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(B),D))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(Aa),D)),aa_real_real(times_times_real(B),C))) ).

% complex_mult
tff(fact_9040_Complex__eq__1,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_complex(complex2(Aa),B) = one_one_complex )
    <=> ( ( Aa = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_9041_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_9042_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_9043_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(plus_plus_real(one_one_real),Xb)),aa_real_real(exp_real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_9044_lemma__exp__total,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y3)
     => ? [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(minus_minus_real(Y3),one_one_real))
          & ( aa_real_real(exp_real,X2) = Y3 ) ) ) ).

% lemma_exp_total
tff(fact_9045_ln__ge__iff,axiom,
    ! [Xb: real,Y3: 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,Y3),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y3)),Xb) ) ) ).

% ln_ge_iff
tff(fact_9046_ln__x__over__x__mono,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Y3)),Y3)),aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Xb)),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_9047_Complex__eq__neg__1,axiom,
    ! [Aa: real,B: real] :
      ( ( aa_real_complex(complex2(Aa),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( Aa = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_9048_real__less__rsqrt,axiom,
    ! [Xb: real,Y3: 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)))),Y3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(sqrt,Y3)) ) ).

% real_less_rsqrt
tff(fact_9049_real__le__rsqrt,axiom,
    ! [Xb: real,Y3: 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)))),Y3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,Y3)) ) ).

% real_le_rsqrt
tff(fact_9050_sqrt__le__D,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_9051_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% exp_le
tff(fact_9052_real__sqrt__unique,axiom,
    ! [Y3: real,Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),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),Y3)
       => ( aa_real_real(sqrt,Xb) = Y3 ) ) ) ).

% real_sqrt_unique
tff(fact_9053_real__le__lsqrt,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),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)),Y3) ) ) ) ).

% real_le_lsqrt
tff(fact_9054_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(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_9055_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y3 )
     => ( Xb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_9056_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Xb )
     => ( Y3 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_9057_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [Aa: 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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Aa),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(plus_plus_real(B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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(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_9058_real__sqrt__sum__squares__ge2,axiom,
    ! [Y3: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_9059_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_9060_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_9061_sqrt__ge__absD,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,Y3))
     => 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)))),Y3) ) ).

% sqrt_ge_absD
tff(fact_9062_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_45
tff(fact_9063_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_45
tff(fact_9064_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% tan_60
tff(fact_9065_real__less__lsqrt,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),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)),Y3) ) ) ) ).

% real_less_lsqrt
tff(fact_9066_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(Xb),Y3)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_9067_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_9068_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Y3: 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(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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_9069_real__sqrt__ge__abs2,axiom,
    ! [Y3: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_9070_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y3))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_9071_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(divide_divide_real(aa_real_real(ln_ln_real,Xb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_9072_arsinh__real__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(Xb),aa_real_real(sqrt,aa_real_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_9073_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_30
tff(fact_9074_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_60
tff(fact_9075_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_9076_arsinh__real__aux,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(Xb),aa_real_real(sqrt,aa_real_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_9077_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(plus_plus_real(aa_real_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_9078_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Y3: 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(times_times_real(aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_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_9079_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(times_times_real(Xb),Y3))),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Y3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_9080_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(powr_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_9081_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_9082_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_9083_cos__x__y__le__one,axiom,
    ! [Xb: real,Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Xb),aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_9084_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y3)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_9085_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(plus_plus_real(Xb),aa_real_real(sqrt,aa_real_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_9086_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(plus_plus_real(one_one_real),aa_real_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_9087_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(minus_minus_real(one_one_real),aa_real_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_9088_cos__arctan,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_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_9089_sin__arctan,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(arctan,Xb)) = aa_real_real(divide_divide_real(Xb),aa_real_real(sqrt,aa_real_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_9090_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(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vd,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_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_9091_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_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,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_9092_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(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),Xb)),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_9093_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(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(times_times_real(aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq2
tff(fact_9094_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(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_9095_arctan__half,axiom,
    ! [Xb: real] : aa_real_real(arctan,Xb) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(Xb),aa_real_real(plus_plus_real(one_one_real),aa_real_real(sqrt,aa_real_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_9096_tanh__real__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))) ).

% tanh_real_altdef
tff(fact_9097_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(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq1
tff(fact_9098_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_9099_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ve,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_9100_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(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_9101_sin__arccos__abs,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y3)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_9102_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_9103_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_9104_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_9105_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_9106_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_9107_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_9108_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_9109_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_9110_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_9111_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_9112_cos__arccos,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y3)) = Y3 ) ) ) ).

% cos_arccos
tff(fact_9113_sin__arcsin,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y3)) = Y3 ) ) ) ).

% sin_arcsin
tff(fact_9114_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_9115_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_9116_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_9117_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A2: complex,R3: real] : Z = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(exp_complex,A2)) ).

% complex_exp_exists
tff(fact_9118_choose__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),one_one_nat) = Nb ).

% choose_one
tff(fact_9119_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_9120_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_9121_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_9122_choose__mult__lemma,axiom,
    ! [Mb: 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,Mb),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),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,Mb),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_9123_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_9124_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_9125_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_9126_Suc__times__binomial__add,axiom,
    ! [Aa: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Aa)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B))),aa_nat_nat(suc,Aa))) = 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,Aa),B))),Aa)) ).

% Suc_times_binomial_add
tff(fact_9127_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_9128_choose__mult,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),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,Mb),K))) ) ) ) ).

% choose_mult
tff(fact_9129_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_9130_arccos__le__arccos,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y3)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_9131_arccos__le__mono,axiom,
    ! [Xb: real,Y3: 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,Y3)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_9132_arccos__eq__iff,axiom,
    ! [Xb: real,Y3: 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,Y3)),one_one_real) )
     => ( ( aa_real_real(arccos,Xb) = aa_real_real(arccos,Y3) )
      <=> ( Xb = Y3 ) ) ) ).

% arccos_eq_iff
tff(fact_9133_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_9134_arcsin__le__arcsin,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y3)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_9135_arcsin__le__mono,axiom,
    ! [Xb: real,Y3: 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,Y3)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ) ).

% arcsin_le_mono
tff(fact_9136_arcsin__eq__iff,axiom,
    ! [Xb: real,Y3: 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,Y3)),one_one_real)
       => ( ( aa_real_real(arcsin,Xb) = aa_real_real(arcsin,Y3) )
        <=> ( Xb = Y3 ) ) ) ) ).

% arcsin_eq_iff
tff(fact_9137_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_9138_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_9139_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_9140_binomial__antimono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_9141_binomial__mono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_mono
tff(fact_9142_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_9143_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_9144_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_9145_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_9146_arccos__lbound,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y3)) ) ) ).

% arccos_lbound
tff(fact_9147_arccos__less__arccos,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y3)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_9148_arccos__less__mono,axiom,
    ! [Xb: real,Y3: 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,Y3)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_9149_arccos__ubound,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y3)),pi) ) ) ).

% arccos_ubound
tff(fact_9150_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_9151_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y3)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9152_arcsin__less__mono,axiom,
    ! [Xb: real,Y3: 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,Y3)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ) ) ).

% arcsin_less_mono
tff(fact_9153_cos__arccos__abs,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y3)) = Y3 ) ) ).

% cos_arccos_abs
tff(fact_9154_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_9155_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_9156_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_9157_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_strict_mono
tff(fact_9158_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_9159_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_9160_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_9161_arccos__lt__bounded,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y3))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y3)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_9162_arccos__bounded,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y3))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y3)),pi) ) ) ) ).

% arccos_bounded
tff(fact_9163_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_9164_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_9165_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(minus_minus_real(pi),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_minus
tff(fact_9166_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_9167_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_9168_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_vf),one_one_real) ).

% power_half_series
tff(fact_9169_arccos,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y3))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y3)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y3)) = Y3 ) ) ) ) ).

% arccos
tff(fact_9170_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(minus_minus_real(pi),aa_real_real(arccos,Xb)) ) ) ).

% arccos_minus_abs
tff(fact_9171_sums__if_H,axiom,
    ! [G: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => aa_real_o(sums_real(aTP_Lamm_vg(G)),Xb) ) ).

% sums_if'
tff(fact_9172_sums__if,axiom,
    ! [G: fun_nat_real,Xb: real,F: fun_nat_real,Y3: real] :
      ( aa_real_o(sums_real(G),Xb)
     => ( aa_real_o(sums_real(F),Y3)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_vh(G),F)),aa_real_real(plus_plus_real(Xb),Y3)) ) ) ).

% sums_if
tff(fact_9173_arccos__le__pi2,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_9174_arcsin__lt__bounded,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9175_arcsin__bounded,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_9176_arcsin__ubound,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_9177_arcsin__lbound,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3)) ) ) ).

% arcsin_lbound
tff(fact_9178_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_9179_cos__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_vi,Xb)),aa_real_real(cos_real,Xb)) ).

% cos_paired
tff(fact_9180_arcsin,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y3)) = Y3 ) ) ) ) ).

% arcsin
tff(fact_9181_arcsin__pi,axiom,
    ! [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y3)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y3)) = Y3 ) ) ) ) ).

% arcsin_pi
tff(fact_9182_arcsin__le__iff,axiom,
    ! [Xb: real,Y3: 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(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),Y3)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sin_real,Y3)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9183_le__arcsin__iff,axiom,
    ! [Xb: real,Y3: 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(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(arcsin,Xb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y3)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9184_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(minus_minus_real(Theta),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,K3)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_9185_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(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_9186_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(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_9187_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_9188_atMost__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% atMost_Suc
tff(fact_9189_finite__nat__iff__bounded__le,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
    <=> ? [K2: nat] : aa_set_nat_o(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_atMost_nat,K2)) ) ).

% finite_nat_iff_bounded_le
tff(fact_9190_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_9191_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vj,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb)) ).

% sum_choose_upper
tff(fact_9192_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vk,R2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_9193_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vl,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),Mb) ).

% choose_rising_sum(2)
tff(fact_9194_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vl,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9195_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vm(Mb),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_9196_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vn(Mb),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),R2) ).

% vandermonde
tff(fact_9197_choose__row__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),aa_nat_set_nat(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_9198_binomial,axiom,
    ! [Aa: 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,Aa),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vo(Aa),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_9199_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9200_polynomial__product__nat,axiom,
    ! [Mb: nat,Aa: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_nat(Aa,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_nat(B,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vp(Aa),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vp(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_vr(Aa),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_9201_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vs,Nb)),aa_nat_set_nat(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_9202_binomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% binomial_r_part_sum
tff(fact_9203_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vt,Nb)),aa_nat_set_nat(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_9204_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_9205_real__scaleR__def,axiom,
    ! [Aa: real,Xb: real] : aa_real_real(real_V1485227260804924795R_real(Aa),Xb) = aa_real_real(times_times_real(Aa),Xb) ).

% real_scaleR_def
tff(fact_9206_complex__scaleR,axiom,
    ! [R2: real,Aa: real,B: real] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),aa_real_complex(complex2(Aa),B)) = aa_real_complex(complex2(aa_real_real(times_times_real(R2),Aa)),aa_real_real(times_times_real(R2),B)) ).

% complex_scaleR
tff(fact_9207_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_9208_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_9209_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(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_9210_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_9211_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_9212_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_9213_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_9214_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_9215_floor__divide__eq__div__numeral,axiom,
    ! [Aa: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,Aa)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Aa)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_9216_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(one_one_real),aa_num_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_9217_floor__minus__divide__eq__div__numeral,axiom,
    ! [Aa: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,Aa)),aa_num_real(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,Aa))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_9218_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_9219_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_9220_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_9221_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_9222_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),aa_num_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_9223_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_9224_complex__i__not__numeral,axiom,
    ! [W: num] : imaginary_unit != numera6690914467698888265omplex(W) ).

% complex_i_not_numeral
tff(fact_9225_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_9226_complex__i__not__neg__numeral,axiom,
    ! [W: num] : imaginary_unit != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_i_not_neg_numeral
tff(fact_9227_Complex__eq__i,axiom,
    ! [Xb: real,Y3: real] :
      ( ( aa_real_complex(complex2(Xb),Y3) = imaginary_unit )
    <=> ( ( Xb = zero_zero_real )
        & ( Y3 = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9228_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9229_i__mult__Complex,axiom,
    ! [Aa: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(complex2(Aa),B)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),Aa) ).

% i_mult_Complex
tff(fact_9230_Complex__mult__i,axiom,
    ! [Aa: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(Aa),B)),imaginary_unit) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),Aa) ).

% Complex_mult_i
tff(fact_9231_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(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_9232_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(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_9233_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(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_9234_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_9235_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(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_9236_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(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_9237_floor__divide__real__eq__div,axiom,
    ! [B: int,Aa: 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(divide_divide_real(Aa),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,Aa)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_9238_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_9239_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_9240_Complex__eq,axiom,
    ! [Aa: real,B: real] : aa_real_complex(complex2(Aa),B) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,Aa)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_9241_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A2: real] : Z = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A2))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A2))))) ).

% complex_split_polar
tff(fact_9242_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(log(B),Xb)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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(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_9243_cmod__unit__one,axiom,
    ! [Aa: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,Aa))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,Aa))))) = one_one_real ).

% cmod_unit_one
tff(fact_9244_cmod__complex__polar,axiom,
    ! [R2: real,Aa: 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,Aa))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,Aa)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_9245_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(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9246_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(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_9247_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_9248_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9249_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_9250_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_9251_csqrt__1,axiom,
    aa_complex_complex(csqrt,one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_9252_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_9253_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_9254_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_9255_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),Mb)) ) ) ).

% fact_div_fact
tff(fact_9256_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_9257_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9258_norm__cis,axiom,
    ! [Aa: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,Aa)) = one_one_real ).

% norm_cis
tff(fact_9259_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9260_cis__pi,axiom,
    aa_real_complex(cis,pi) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cis_pi
tff(fact_9261_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9262_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9263_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_9264_divide__real__def,axiom,
    ! [Xb: real,Y3: real] : aa_real_real(divide_divide_real(Xb),Y3) = aa_real_real(times_times_real(Xb),aa_real_real(inverse_inverse_real,Y3)) ).

% divide_real_def
tff(fact_9265_cis__mult,axiom,
    ! [Aa: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(cis,Aa)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(plus_plus_real(Aa),B)) ).

% cis_mult
tff(fact_9266_inverse__powr,axiom,
    ! [Y3: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
     => ( aa_real_real(powr_real(aa_real_real(inverse_inverse_real,Y3)),Aa) = aa_real_real(inverse_inverse_real,aa_real_real(powr_real(Y3),Aa)) ) ) ).

% inverse_powr
tff(fact_9267_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,J)) = groups1705073143266064639nt_int(aTP_Lamm_aj,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_9268_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D3: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D3),E2)
         => ( aa_real_o(P,D3)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( 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_9269_DeMoivre,axiom,
    ! [Aa: real,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,Aa)),Nb) = aa_real_complex(cis,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),Aa)) ).

% DeMoivre
tff(fact_9270_forall__pos__mono,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D3: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D3),E2)
         => ( aa_real_o(P,D3)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono
tff(fact_9271_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))),E) ) ) ).

% real_arch_inverse
tff(fact_9272_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(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_9273_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_aj,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_9274_log__inverse,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(log(Aa),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(log(Aa),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_9275_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_9276_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(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_9277_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(plus_plus_real(Xb),aa_real_real(inverse_inverse_real,Xb))) ) ).

% plus_inverse_ge_2
tff(fact_9278_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_9279_tan__cot,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(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_9280_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(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_9281_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(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_9282_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(minus_minus_real(aa_real_real(sin_real,Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vc,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_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_9283_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(aa_nat6454746962644518025omplex(aTP_Lamm_vu,Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_do,Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9284_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),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_9285_sinh__real__le__iff,axiom,
    ! [Xb: real,Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ).

% sinh_real_le_iff
tff(fact_9286_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_9287_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_9288_cot__npi,axiom,
    ! [Nb: nat] : aa_real_real(cot_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_9289_cot__periodic,axiom,
    ! [Xb: real] : aa_real_real(cot_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xb) ).

% cot_periodic
tff(fact_9290_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_9291_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_9292_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9293_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Y3: 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,Y3),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,Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9294_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_9295_divide__complex__def,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),aa_complex_complex(invers8013647133539491842omplex,Y3)) ).

% divide_complex_def
tff(fact_9296_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y3: 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,Y3),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,Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9297_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y3))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9298_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y3)) ) ) ).

% cosh_real_strict_mono
tff(fact_9299_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_9300_complex__inverse,axiom,
    ! [Aa: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(Aa),B)) = aa_real_complex(complex2(aa_real_real(divide_divide_real(Aa),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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(divide_divide_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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_9301_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(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9302_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xb)) ) ) ).

% cot_gt_zero
tff(fact_9303_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(divide_divide_real(aa_real_real(minus_minus_real(Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9304_tan__cot_H,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) = aa_real_real(cot_real,Xb) ).

% tan_cot'
tff(fact_9305_arctan__def,axiom,
    ! [Y3: real] : aa_real_real(arctan,Y3) = the_real(aa_real_fun_real_o(aTP_Lamm_vv,Y3)) ).

% arctan_def
tff(fact_9306_arcsin__def,axiom,
    ! [Y3: real] : aa_real_real(arcsin,Y3) = the_real(aa_real_fun_real_o(aTP_Lamm_vw,Y3)) ).

% arcsin_def
tff(fact_9307_modulo__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,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,Mb)),
        $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,Mb),Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))))) ) ).

% modulo_int_unfold
tff(fact_9308_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_9309_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_9310_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_9311_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_9312_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_9313_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_9314_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L3: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_9315_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_9316_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_9317_ln__real__def,axiom,
    ! [Xb: real] : aa_real_real(ln_ln_real,Xb) = the_real(aa_real_fun_real_o(aTP_Lamm_vx,Xb)) ).

% ln_real_def
tff(fact_9318_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_9319_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_9320_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_bb) ) ) ).

% ln_neg_is_const
tff(fact_9321_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_9322_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_9323_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_9324_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_9325_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_9326_arccos__def,axiom,
    ! [Y3: real] : aa_real_real(arccos,Y3) = the_real(aa_real_fun_real_o(aTP_Lamm_vy,Y3)) ).

% arccos_def
tff(fact_9327_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_9328_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_9329_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_9330_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_9331_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K2: int] :
            ( ( A1 = K2 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K2) ) )
        | ? [L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R5: int,L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9332_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) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9333_pi__half,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_vz) ).

% pi_half
tff(fact_9334_pi__def,axiom,
    pi = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_vz)) ).

% pi_def
tff(fact_9335_divide__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,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,Mb),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_9336_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_9337_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_9338_sgn__real__def,axiom,
    ! [Aa: real] :
      aa_real_real(sgn_sgn_real,Aa) = $ite(
        Aa = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_9339_sgn__power__injE,axiom,
    ! [Aa: real,Nb: nat,Xb: real,B: real] :
      ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Aa)),Nb)) = Xb )
     => ( ( Xb = aa_real_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)
         => ( Aa = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9340_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_9341_floor__real__def,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aa_real_fun_int_o(aTP_Lamm_wa,Xb)) ).

% floor_real_def
tff(fact_9342_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_9343_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),Xb)) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Xb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_9344_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(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(divide_divide_real(aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_do,Nb))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_wb(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9345_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_9346_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_9347_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(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(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_9348_real__root__zero,axiom,
    ! [Nb: nat] : aa_real_real(aa_nat_fun_real_real(root,Nb),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9349_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_9350_real__root__Suc__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xb) = Xb ).

% real_root_Suc_0
tff(fact_9351_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_nat_fun_real_real(root,Nb),Y3) )
      <=> ( Xb = Y3 ) ) ) ).

% real_root_eq_iff
tff(fact_9352_root__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xb) = zero_zero_real ).

% root_0
tff(fact_9353_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9354_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9355_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y3) ) ) ).

% real_root_less_iff
tff(fact_9356_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y3) ) ) ).

% real_root_le_iff
tff(fact_9357_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_9358_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_9359_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_9360_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9361_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9362_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_9363_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_9364_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9365_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) ).

% real_root_gt_0_iff
tff(fact_9366_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9367_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3) ) ) ).

% real_root_ge_0_iff
tff(fact_9368_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_9369_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9370_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y3) ) ) ).

% real_root_gt_1_iff
tff(fact_9371_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y3) ) ) ).

% real_root_ge_1_iff
tff(fact_9372_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9373_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_9374_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y3: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_int_nat(nat2,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y3 ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_9375_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,Xb: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y3 = 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_9376_nat__ceiling__le__eq,axiom,
    ! [Xb: real,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Aa)) ) ).

% nat_ceiling_le_eq
tff(fact_9377_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_9378_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_9379_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_9380_nat__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,Aa)),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,Aa),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_9381_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: 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,Aa))
    <=> 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) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9382_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Aa: 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,Aa))
    <=> 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) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9383_nat__le__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Aa)),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,Aa),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_9384_real__root__divide,axiom,
    ! [Nb: nat,Xb: real,Y3: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(divide_divide_real(Xb),Y3)) = aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3)) ).

% real_root_divide
tff(fact_9385_real__root__mult__exp,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Xb) = aa_real_real(aa_nat_fun_real_real(root,Mb),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_mult_exp
tff(fact_9386_real__root__mult,axiom,
    ! [Nb: nat,Xb: real,Y3: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(times_times_real(Xb),Y3)) = aa_real_real(times_times_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3)) ).

% real_root_mult
tff(fact_9387_real__root__minus,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_minus
tff(fact_9388_real__root__commute,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,Mb),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_nat_fun_real_real(root,Mb),Xb)) ).

% real_root_commute
tff(fact_9389_real__root__inverse,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_inverse
tff(fact_9390_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_9391_real__root__pos__pos__le,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_9392_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_9393_nat__mono,axiom,
    ! [Xb: int,Y3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y3)) ) ).

% nat_mono
tff(fact_9394_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_9395_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_9396_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_13: nat] : aa_nat_o(P,X_13)
    <=> ! [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_9397_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
    <=> ? [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_9398_unset__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% unset_bit_nat_def
tff(fact_9399_nat__mask__eq,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% nat_mask_eq
tff(fact_9400_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3)) ) ) ).

% real_root_less_mono
tff(fact_9401_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y3: 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),Y3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y3)) ) ) ).

% real_root_le_mono
tff(fact_9402_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(aa_nat_fun_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(aa_nat_fun_real_real(root,Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_9403_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_9404_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_9405_zless__nat__eq__int__zless,axiom,
    ! [Mb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_9406_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_9407_int__eq__iff,axiom,
    ! [Mb: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = Z )
    <=> ( ( Mb = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_9408_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_9409_nat__int__add,axiom,
    ! [Aa: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B) ).

% nat_int_add
tff(fact_9410_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_9411_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_9412_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_9413_and__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% and_nat_def
tff(fact_9414_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_9415_or__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% or_nat_def
tff(fact_9416_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_9417_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_9418_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_9419_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_9420_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N3: 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),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N3),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9421_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9422_root__abs__power,axiom,
    ! [Nb: nat,Y3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),Nb))) = aa_real_real(abs_abs_real,Y3) ) ) ).

% root_abs_power
tff(fact_9423_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_9424_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_9425_nat__eq__iff2,axiom,
    ! [Mb: nat,W: int] :
      ( ( Mb = 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,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_9426_nat__eq__iff,axiom,
    ! [W: int,Mb: nat] :
      ( ( aa_int_nat(nat2,W) = Mb )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_9427_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_9428_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_9429_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_9430_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_9431_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_9432_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_9433_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y3: 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),Y3)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y3)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9434_nat__div__distrib_H,axiom,
    ! [Y3: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y3)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y3)) ) ) ).

% nat_div_distrib'
tff(fact_9435_nat__div__distrib,axiom,
    ! [Xb: int,Y3: 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),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y3)) ) ) ).

% nat_div_distrib
tff(fact_9436_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_9437_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_9438_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_9439_nat__mod__distrib,axiom,
    ! [Xb: int,Y3: 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),Y3)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y3)) ) ) ) ).

% nat_mod_distrib
tff(fact_9440_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_9441_le__nat__floor,axiom,
    ! [Xb: nat,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Aa))) ) ).

% le_nat_floor
tff(fact_9442_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_9443_real__root__strict__increasing,axiom,
    ! [Nb: nat,N3: 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),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N3),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9444_real__root__decreasing,axiom,
    ! [Nb: nat,N3: 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),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N3),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_9445_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(aa_nat_fun_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_9446_odd__real__root__unique,axiom,
    ! [Nb: nat,Y3: 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,Y3),Nb) = Xb )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y3 ) ) ) ).

% odd_real_root_unique
tff(fact_9447_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(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ).

% odd_real_root_pow
tff(fact_9448_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_9449_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_9450_real__root__pos__unique,axiom,
    ! [Nb: nat,Y3: 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),Y3)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),Nb) = Xb )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y3 ) ) ) ) ).

% real_root_pos_unique
tff(fact_9451_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_9452_root__sgn__power,axiom,
    ! [Nb: nat,Y3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(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))) = Y3 ) ) ).

% root_sgn_power
tff(fact_9453_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(times_times_real(aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_9454_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_9455_nat__less__iff,axiom,
    ! [W: int,Mb: 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)),Mb)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,Mb)) ) ) ).

% nat_less_iff
tff(fact_9456_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_9457_nat__abs__int__diff,axiom,
    ! [Aa: 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,Aa)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),B)) ).

% nat_abs_int_diff
tff(fact_9458_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_9459_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_9460_real__root__increasing,axiom,
    ! [Nb: nat,N3: 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),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N3),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_9461_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xb: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_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)
         => ! [Y4: real] :
              ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y4)),Nb)) = Xb )
             => aa_real_o(P,Y4) ) ) ) ) ).

% split_root
tff(fact_9462_nat__dvd__iff,axiom,
    ! [Z: int,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),Mb)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb)),Mb = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_9463_log__root,axiom,
    ! [Nb: nat,Aa: 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),Aa)
       => ( aa_real_real(log(B),aa_real_real(aa_nat_fun_real_real(root,Nb),Aa)) = aa_real_real(divide_divide_real(aa_real_real(log(B),Aa)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9464_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(log(aa_real_real(aa_nat_fun_real_real(root,Nb),B)),Xb) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Xb)) ) ) ) ).

% log_base_root
tff(fact_9465_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9466_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(powr_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9467_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_9468_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(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_9469_floor__rat__def,axiom,
    ! [Xb: rat] : aa_rat_int(archim3151403230148437115or_rat,Xb) = the_int(aTP_Lamm_wc(Xb)) ).

% floor_rat_def
tff(fact_9470_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_wd(Z))) ).

% Arg_def
tff(fact_9471_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_9472_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),zero_zero_int) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) ).

% concat_bit_of_zero_2
tff(fact_9473_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_9474_sgn__rat__def,axiom,
    ! [Aa: rat] :
      aa_rat_rat(sgn_sgn_rat,Aa) = $ite(
        Aa = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9475_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_9476_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_9477_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S2)
           => ! [T3: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T3)
               => ( R2 != aa_rat_rat(plus_plus_rat(S2),T3) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9478_less__eq__rat__def,axiom,
    ! [Xb: rat,Y3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y3)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y3)
        | ( Xb = Y3 ) ) ) ).

% less_eq_rat_def
tff(fact_9479_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9480_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_9481_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_9482_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B: int] : aa_int_fun_int_int(bit_concat_bit(Nb),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),B)) = aa_int_fun_int_int(bit_concat_bit(Nb),B) ).

% concat_bit_take_bit_eq
tff(fact_9483_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K: int,L: int,R2: int,S: int] :
      ( ( 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(Nb),R2),S) )
    <=> ( ( 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),R2) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_9484_take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(uminus_uminus_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),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_9485_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_9486_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Mb),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9487_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_9488_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_9489_not__take__bit__negative,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)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_9490_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_se2923211474154528505it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_9491_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_9492_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_9493_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_9494_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),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_9495_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9496_take__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_9497_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_9498_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_9499_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9500_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_9501_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_9502_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_9503_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_9504_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_9505_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_9506_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_9507_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_9508_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_9509_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_9510_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_9511_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_9512_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_9513_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_9514_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_9515_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_9516_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_9517_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_9518_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_9519_signed__take__bit__negative__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)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_9520_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_9521_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_9522_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_9523_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_9524_add__inc,axiom,
    ! [Xb: num,Y3: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),aa_num_num(inc,Y3)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),Y3)) ).

% add_inc
tff(fact_9525_bit__or__int__iff,axiom,
    ! [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_se1409905431419307370or_int,K),L)),Nb)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
        | aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_9526_bit__and__int__iff,axiom,
    ! [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_se725231765392027082nd_int,K),L)),Nb)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_9527_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_9528_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9529_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_9530_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_9531_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_9532_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_9533_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_9534_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_9535_mult__inc,axiom,
    ! [Xb: num,Y3: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Xb),aa_num_num(inc,Y3)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Y3)),Xb) ).

% mult_inc
tff(fact_9536_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,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,Mb),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9537_bit__concat__bit__iff,axiom,
    ! [Mb: 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(Mb),K),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9538_signed__take__bit__eq__concat__bit,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(bit_concat_bit(Nb),K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_9539_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_9540_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_9541_set__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))) ).

% set_bit_eq
tff(fact_9542_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_9543_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_9544_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9545_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_9546_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_9547_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),one) = Xb ).

% pow.simps(1)
tff(fact_9548_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_9549_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_9550_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,aa_real_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_9551_bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),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_9552_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(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9553_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(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9554_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_we),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9555_setceilmax,axiom,
    ! [S: vEBT_VEBT,Mb: nat,Listy: list_VEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(S),Mb)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Listy))
           => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
       => ( ( Mb = 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(log(aa_num_real(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,S)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(S,set_VEBT_VEBT2(Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_9556_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)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summarya,set_VEBT_VEBT2(TreeLista))))) ) ).

% height_compose_list
tff(fact_9557_max__ins__scaled,axiom,
    ! [Nb: nat,X14: vEBT_VEBT,Mb: nat,X13: 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,X14))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(lattic8265883725875713057ax_nat,insert_nat(aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14),image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))))) ).

% max_ins_scaled
tff(fact_9558_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13,I))),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Foo),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))) ) ).

% height_i_max
tff(fact_9559_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Nb: nat,X14: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
     => 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(X13,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,X14)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))))) ) ).

% max_idx_list
tff(fact_9560_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gd,Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_9561_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9562_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_9563_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_9564_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% quotient_of_number(5)
tff(fact_9565_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_9566_divide__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa_rat_rat(divide_divide_rat(Q2),R2) = aa_rat_rat(times_times_rat(Q2),aa_rat_rat(inverse_inverse_rat,R2)) ).

% divide_rat_def
tff(fact_9567_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Uu,Dega,TreeLista,Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summarya,set_VEBT_VEBT2(TreeLista))))) ).

% VEBT_internal.height.simps(2)
tff(fact_9568_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xb) = Y3 )
     => ( ( ? [A2: $o,B3: $o] : Xb = vEBT_Leaf((A2),(B3))
         => ( Y3 != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Uu2,Deg,TreeList,Summary) )
             => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_9569_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wf(Mb),Nb)))) ).

% divide_nat_def
tff(fact_9570_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wg),quotient_of(P2)) ).

% rat_abs_code
tff(fact_9571_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wh),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9572_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_wj(Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9573_rat__floor__code,axiom,
    ! [P2: rat] : aa_rat_int(archim3151403230148437115or_rat,P2) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(divide_divide_int),quotient_of(P2)) ).

% rat_floor_code
tff(fact_9574_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_wl(Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9575_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xb) = Y3 )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),Xb)
       => ( ! [A2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B3)) )
             => ( ( Y3 = zero_zero_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf((A2),(B3))) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,Deg,TreeList,Summary) )
               => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList))))) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Node(Uu2,Deg,TreeList,Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_9576_quotient__of__int,axiom,
    ! [Aa: int] : quotient_of(aa_int_rat(of_int,Aa)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Aa),one_one_int) ).

% quotient_of_int
tff(fact_9577_bij__betw__Suc,axiom,
    ! [M6: set_nat,N3: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M6),N3)
    <=> ( image_nat_nat(suc,M6) = N3 ) ) ).

% bij_betw_Suc
tff(fact_9578_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_9579_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_gf,Nb))) = aa_int_int(abs_abs_int,Nb) ) ) ).

% Max_divisors_self_int
tff(fact_9580_zero__notin__Suc__image,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A)) ).

% zero_notin_Suc_image
tff(fact_9581_finite__int__iff__bounded,axiom,
    ! [S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
    <=> ? [K2: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S3)),aa_int_set_int(set_ord_lessThan_int,K2)) ) ).

% finite_int_iff_bounded
tff(fact_9582_finite__int__iff__bounded__le,axiom,
    ! [S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
    <=> ? [K2: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S3)),set_ord_atMost_int(K2)) ) ).

% finite_int_iff_bounded_le
tff(fact_9583_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = set_or1269000886237332187st_nat(one_one_nat,Nb) ).

% image_Suc_lessThan
tff(fact_9584_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9585_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_9586_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9587_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_9588_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9589_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_wn(Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9590_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wp(Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9591_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_9592_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != 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),S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9593_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_9594_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int)) = aa_num_rat(numeral_numeral_rat,K) ).

% Frct_code_post(4)
tff(fact_9595_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(divide_divide_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9596_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wr(Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9597_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_wt(Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9598_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_9599_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_9600_xor__nat__numerals_I1_J,axiom,
    ! [Y3: 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,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% xor_nat_numerals(1)
tff(fact_9601_xor__nat__numerals_I2_J,axiom,
    ! [Y3: 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,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3)) ).

% xor_nat_numerals(2)
tff(fact_9602_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_9603_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_9604_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9605_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          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))),Mb) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9606_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_9607_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M5)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(X5,M5)),aa_nat_real(X5,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9608_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_9609_push__bit__negative__int__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_se545348938243370406it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9610_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),zero_zero_int),L) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),L) ).

% concat_bit_of_zero_1
tff(fact_9611_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_9612_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_9613_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_9614_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_9615_bit__xor__int__iff,axiom,
    ! [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_se6526347334894502574or_int,K),L)),Nb)
    <=> ~ ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
        <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_9616_push__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K)) ).

% push_bit_nat_eq
tff(fact_9617_XOR__lower,axiom,
    ! [Xb: int,Y3: 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),Y3)
       => 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),Y3)) ) ) ).

% XOR_lower
tff(fact_9618_set__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9619_flip__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9620_bit__push__bit__iff__int,axiom,
    ! [Mb: 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,Mb),K)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9621_xor__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% xor_nat_def
tff(fact_9622_bit__push__bit__iff__nat,axiom,
    ! [Mb: 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,Mb),Q2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9623_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_9624_concat__bit__def,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(bit_se1409905431419307370or_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_def
tff(fact_9625_set__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)) ).

% set_bit_int_def
tff(fact_9626_push__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_9627_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_9628_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_9629_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Y3: 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,Y3),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),Y3)),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_9630_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_9631_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_9632_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or4665077453230672383an_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9633_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_9634_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_9635_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_9636_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_9637_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : set_or4665077453230672383an_nat(Mb,aa_nat_nat(suc,Mb)) = insert_nat(Mb,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9638_and__minus__minus__numerals,axiom,
    ! [Mb: 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,Mb))),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,Mb)),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_9639_or__minus__minus__numerals,axiom,
    ! [Mb: 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,Mb))),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,Mb)),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_9640_bit__not__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),Nb)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% bit_not_int_iff
tff(fact_9641_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [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_9642_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [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_9643_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_9644_or__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9645_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_9646_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_9647_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_9648_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_9649_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_9650_xor__int__def,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(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),L)) ).

% xor_int_def
tff(fact_9651_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N3),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9652_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_9653_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_9654_and__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(4)
tff(fact_9655_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_9656_or__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9657_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_9658_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_9659_int__numeral__or__not__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = 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,Mb),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9660_int__numeral__not__or__num__neg,axiom,
    ! [Mb: 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,Mb))),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),Mb))) ).

% int_numeral_not_or_num_neg
tff(fact_9661_numeral__or__not__num__eq,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9662_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      set_or4665077453230672383an_nat(Mb,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),insert_nat(Nb,set_or4665077453230672383an_nat(Mb,Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9663_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_9664_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_9665_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_9666_and__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9667_and__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(7)
tff(fact_9668_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_9669_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_9670_or__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9671_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K: num] :
      set_or4665077453230672383an_nat(Mb,aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),set_or4665077453230672383an_nat(Mb,aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9672_and__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9673_and__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(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,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9674_or__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(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,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9675_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9676_or__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9677_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xb: nat,Y3: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wu,C),set_or4665077453230672383an_nat(Xb,Y3)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y3),
        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,Y3),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y3),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9678_and__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9679_or__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9680_or__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9681_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_9682_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,Aa: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Aa,I2)),aa_nat_nat(Aa,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_wv(Aa),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(Aa),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_9683_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_9684_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_ww,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_9685_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9686_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_9687_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_9688_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
    <=> aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq
tff(fact_9689_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),D)
     => aa_nat_o(vEBT_VEBT_valid(Ta),D) ) ).

% valid_eq1
tff(fact_9690_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
     => aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq2
tff(fact_9691_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9692_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9693_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
tff(fact_9694_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9695_complex__Re__numeral,axiom,
    ! [V: num] : aa_complex_real(re,numera6690914467698888265omplex(V)) = aa_num_real(numeral_numeral_real,V) ).

% complex_Re_numeral
tff(fact_9696_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(aa_complex_real(re,Z)),aa_num_real(numeral_numeral_real,W)) ).

% Re_divide_numeral
tff(fact_9697_sums__Re,axiom,
    ! [X5: fun_nat_complex,Aa: complex] :
      ( aa_complex_o(sums_complex(X5),Aa)
     => aa_real_o(sums_real(aTP_Lamm_wx(X5)),aa_complex_real(re,Aa)) ) ).

% sums_Re
tff(fact_9698_Cauchy__Re,axiom,
    ! [X5: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X5)
     => topolo4055970368930404560y_real(aTP_Lamm_wx(X5)) ) ).

% Cauchy_Re
tff(fact_9699_complex__Re__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% complex_Re_le_cmod
tff(fact_9700_one__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,one_one_complex) = one_one_real ).

% one_complex.simps(1)
tff(fact_9701_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(re,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb)) = aa_real_real(times_times_real(R2),aa_complex_real(re,Xb)) ).

% scaleR_complex.simps(1)
tff(fact_9702_summable__Re,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_real(aTP_Lamm_wx(F)) ) ).

% summable_Re
tff(fact_9703_abs__Re__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% abs_Re_le_cmod
tff(fact_9704_Re__csqrt,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z))) ).

% Re_csqrt
tff(fact_9705_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real)
    <=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_9706_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,Aa: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),Aa)) = aa_complex_real(re,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,Aa)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_9707_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),
        aa_real_real(times_times_real($ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9708_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(times_times_real($ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9709_csqrt__of__real__nonpos,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xb) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_9710_Im__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_complex_real(im,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_9711_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_9712_Im__i__times,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Z)) = aa_complex_real(re,Z) ).

% Im_i_times
tff(fact_9713_Re__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),Nb) ) ) ).

% Re_power_real
tff(fact_9714_Re__i__times,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Z)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z)) ).

% Re_i_times
tff(fact_9715_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(aa_complex_real(im,Z)),aa_num_real(numeral_numeral_real,W)) ).

% Im_divide_numeral
tff(fact_9716_csqrt__of__real__nonneg,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb))
       => ( aa_complex_complex(csqrt,Xb) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,Xb))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_9717_csqrt__minus,axiom,
    ! [Xb: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xb)),zero_zero_real)
        | ( ( aa_complex_real(im,Xb) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_complex_complex(csqrt,Xb)) ) ) ).

% csqrt_minus
tff(fact_9718_sums__Im,axiom,
    ! [X5: fun_nat_complex,Aa: complex] :
      ( aa_complex_o(sums_complex(X5),Aa)
     => aa_real_o(sums_real(aTP_Lamm_wy(X5)),aa_complex_real(im,Aa)) ) ).

% sums_Im
tff(fact_9719_Cauchy__Im,axiom,
    ! [X5: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X5)
     => topolo4055970368930404560y_real(aTP_Lamm_wy(X5)) ) ).

% Cauchy_Im
tff(fact_9720_imaginary__unit_Osimps_I2_J,axiom,
    aa_complex_real(im,imaginary_unit) = one_one_real ).

% imaginary_unit.simps(2)
tff(fact_9721_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_9722_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(im,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb)) = aa_real_real(times_times_real(R2),aa_complex_real(im,Xb)) ).

% scaleR_complex.simps(2)
tff(fact_9723_sums__complex__iff,axiom,
    ! [F: fun_nat_complex,Xb: complex] :
      ( aa_complex_o(sums_complex(F),Xb)
    <=> ( aa_real_o(sums_real(aTP_Lamm_wx(F)),aa_complex_real(re,Xb))
        & aa_real_o(sums_real(aTP_Lamm_wy(F)),aa_complex_real(im,Xb)) ) ) ).

% sums_complex_iff
tff(fact_9724_summable__Im,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_real(aTP_Lamm_wy(F)) ) ).

% summable_Im
tff(fact_9725_abs__Im__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% abs_Im_le_cmod
tff(fact_9726_summable__complex__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
    <=> ( summable_real(aTP_Lamm_wx(F))
        & summable_real(aTP_Lamm_wy(F)) ) ) ).

% summable_complex_iff
tff(fact_9727_times__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y3)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(im,Y3))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(re,Y3))) ).

% times_complex.simps(2)
tff(fact_9728_cmod__Re__le__iff,axiom,
    ! [Xb: complex,Y3: complex] :
      ( ( aa_complex_real(im,Xb) = aa_complex_real(im,Y3) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y3))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))),aa_real_real(abs_abs_real,aa_complex_real(re,Y3))) ) ) ).

% cmod_Re_le_iff
tff(fact_9729_cmod__Im__le__iff,axiom,
    ! [Xb: complex,Y3: complex] :
      ( ( aa_complex_real(re,Xb) = aa_complex_real(re,Y3) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y3))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xb))),aa_real_real(abs_abs_real,aa_complex_real(im,Y3))) ) ) ).

% cmod_Im_le_iff
tff(fact_9730_times__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y3)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(re,Y3))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(im,Y3))) ).

% times_complex.simps(1)
tff(fact_9731_scaleR__complex_Ocode,axiom,
    ! [R2: real,Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb) = aa_real_complex(complex2(aa_real_real(times_times_real(R2),aa_complex_real(re,Xb))),aa_real_real(times_times_real(R2),aa_complex_real(im,Xb))) ).

% scaleR_complex.code
tff(fact_9732_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z)) = zero_zero_real )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z))) ) ) ).

% csqrt_principal
tff(fact_9733_cmod__le,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))) ).

% cmod_le
tff(fact_9734_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,Aa: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),Aa)) = aa_complex_real(im,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,Aa)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_9735_Re__exp,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(times_times_real(aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).

% Re_exp
tff(fact_9736_Im__exp,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(times_times_real(aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).

% Im_exp
tff(fact_9737_complex__eq,axiom,
    ! [Aa: complex] : Aa = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,Aa))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,Aa)))) ).

% complex_eq
tff(fact_9738_times__complex_Ocode,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y3) = aa_real_complex(complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(re,Y3))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(im,Y3)))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(im,Y3))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(re,Y3)))) ).

% times_complex.code
tff(fact_9739_exp__eq__polar,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,Z) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Z)))),aa_real_complex(cis,aa_complex_real(im,Z))) ).

% exp_eq_polar
tff(fact_9740_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9741_Im__power2,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Xb))),aa_complex_real(im,Xb)) ).

% Im_power2
tff(fact_9742_Re__power2,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9743_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9744_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9745_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(divide_divide_real(aa_complex_real(re,Xb)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9746_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% complex_neq_0
tff(fact_9747_Re__divide,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(re,Y3))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(im,Y3)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9748_csqrt__square,axiom,
    ! [B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_9749_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_9750_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9751_Im__divide,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(re,Y3))),aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(im,Y3)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9752_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(times_times_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% complex_abs_le_norm
tff(fact_9753_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9754_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : aa_complex_complex(invers8013647133539491842omplex,Xb) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_complex_real(re,Xb)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9755_Complex__divide,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y3) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(re,Y3))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(im,Y3)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(re,Y3))),aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(im,Y3)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9756_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,R2),Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_complex_real(re,R2))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9757_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,R2),Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_complex_real(re,R2)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9758_real__eq__imaginary__iff,axiom,
    ! [Y3: complex,Xb: complex] :
      ( aa_set_complex_o(member_complex(Y3),real_V2521375963428798218omplex)
     => ( aa_set_complex_o(member_complex(Xb),real_V2521375963428798218omplex)
       => ( ( Xb = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Y3) )
        <=> ( ( Xb = zero_zero_complex )
            & ( Y3 = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9759_imaginary__eq__real__iff,axiom,
    ! [Y3: complex,Xb: complex] :
      ( aa_set_complex_o(member_complex(Y3),real_V2521375963428798218omplex)
     => ( aa_set_complex_o(member_complex(Xb),real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Y3) = Xb )
        <=> ( ( Xb = zero_zero_complex )
            & ( Y3 = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9760_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9761_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_hd(L),Qr) ).

% divmod_step_integer_def
tff(fact_9762_complex__cnj__mult,axiom,
    ! [Xb: complex,Y3: complex] : aa_complex_complex(cnj,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y3)) ).

% complex_cnj_mult
tff(fact_9763_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(cnj,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
tff(fact_9764_complex__cnj__one,axiom,
    aa_complex_complex(cnj,one_one_complex) = one_one_complex ).

% complex_cnj_one
tff(fact_9765_complex__cnj__power,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(cnj,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cnj,Xb)),Nb) ).

% complex_cnj_power
tff(fact_9766_complex__cnj__numeral,axiom,
    ! [W: num] : aa_complex_complex(cnj,numera6690914467698888265omplex(W)) = numera6690914467698888265omplex(W) ).

% complex_cnj_numeral
tff(fact_9767_complex__cnj__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(cnj,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_cnj_neg_numeral
tff(fact_9768_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_9769_divmod__integer_H__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% divmod_integer'_def
tff(fact_9770_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_9771_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_9772_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_9773_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_9774_sums__cnj,axiom,
    ! [F: fun_nat_complex,L: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_wz(F)),aa_complex_complex(cnj,L))
    <=> aa_complex_o(sums_complex(F),L) ) ).

% sums_cnj
tff(fact_9775_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_9776_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_9777_Re__complex__div__eq__0,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_9778_Im__complex__div__eq__0,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_9779_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_9780_Re__complex__div__lt__0,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_9781_Re__complex__div__gt__0,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_gt_0
tff(fact_9782_Re__complex__div__ge__0,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_ge_0
tff(fact_9783_Re__complex__div__le__0,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_9784_Im__complex__div__gt__0,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_gt_0
tff(fact_9785_Im__complex__div__lt__0,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_9786_Im__complex__div__ge__0,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_ge_0
tff(fact_9787_Im__complex__div__le__0,axiom,
    ! [Aa: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_9788_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9789_complex__div__gt__0,axiom,
    ! [Aa: complex,B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B)))) ) ) ).

% complex_div_gt_0
tff(fact_9790_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_9791_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_9792_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9793_complex__div__cnj,axiom,
    ! [Aa: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Aa),B) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Aa),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_9794_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,W))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_9795_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,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9796_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
tff(fact_9797_integer__of__num_I3_J,axiom,
    ! [Nb: num] :
      aa_num_Code_integer(code_integer_of_num,aa_num_num(bit1,Nb)) = $let(
        k: code_integer,
        k:= aa_num_Code_integer(code_integer_of_num,Nb),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(k,k),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_9798_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_9799_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_9800_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_9801_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
tff(fact_9802_integer__of__num__triv_I1_J,axiom,
    aa_num_Code_integer(code_integer_of_num,one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9803_integer__of__num_I2_J,axiom,
    ! [Nb: num] :
      aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,Nb)) = $let(
        k: code_integer,
        k:= aa_num_Code_integer(code_integer_of_num,Nb),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_9804_integer__of__num__triv_I2_J,axiom,
    aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,one)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9805_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_xa,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9806_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(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_9807_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_xb,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9808_int__of__integer__max,axiom,
    ! [K: code_integer,L: code_integer] : code_int_of_integer(ord_max_Code_integer(K,L)) = aa_int_int(aa_int_fun_int_int(ord_max_int,code_int_of_integer(K)),code_int_of_integer(L)) ).

% int_of_integer_max
tff(fact_9809_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_9810_times__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(times_times_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% times_integer.rep_eq
tff(fact_9811_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_9812_less__eq__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_9813_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,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_9814_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_xc(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9815_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_xd,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9816_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dj,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_9817_card__atMost,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_9818_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_di,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_9819_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or1269000886237332187st_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_9820_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa_nat_Code_integer(semiri4939895301339042750nteger,code_nat_of_integer(K)) = ord_max_Code_integer(zero_z3403309356797280102nteger,K) ).

% of_nat_of_integer
tff(fact_9821_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9822_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_9823_card__less__Suc2,axiom,
    ! [M6: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xe(M6),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xf(M6),I))) ) ) ).

% card_less_Suc2
tff(fact_9824_card__less__Suc,axiom,
    ! [M6: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xe(M6),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xf(M6),I))) ) ) ).

% card_less_Suc
tff(fact_9825_card__less,axiom,
    ! [M6: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xf(M6),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9826_subset__card__intvl__is__intvl,axiom,
    ! [A: set_nat,K: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A))))
     => ( A = set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9827_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : 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_9828_nat__of__integer__code__post_I2_J,axiom,
    code_nat_of_integer(one_one_Code_integer) = one_one_nat ).

% nat_of_integer_code_post(2)
tff(fact_9829_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S3)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S3))) ) ).

% card_le_Suc_Max
tff(fact_9830_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N3),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N3)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9831_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or4665077453230672383an_nat(zero_zero_nat,aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),S3)) ).

% card_sum_le_nat_sum
tff(fact_9832_card__nth__roots,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_wb(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_9833_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_do,Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_9834_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_xg(L),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_xh(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9835_binomial__def,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_xi(Nb),K))) ).

% binomial_def
tff(fact_9836_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_9837_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_az),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9838_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_xj),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9839_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9840_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_xk,Nb)),Mb) ).

% max_Suc1
tff(fact_9841_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_xl,Nb)),Mb) ).

% max_Suc2
tff(fact_9842_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ai),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ).

% diff_Suc
tff(fact_9843_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_9844_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_9845_drop__bit__negative__int__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_se8568078237143864401it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9846_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_9847_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_9848_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_9849_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_9850_drop__bit__push__bit__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Mb),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,Mb),Nb)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),K)) ).

% drop_bit_push_bit_int
tff(fact_9851_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_9852_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ai),Nat) ).

% pred_def
tff(fact_9853_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_9854_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y3: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,Xb),Xaa) = Y3 )
     => ( Y3 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,aa_nat_nat(suc,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9855_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Mb: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,K),Mb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),K),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),Mb)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,K)))) ).

% prod_decode_aux.simps
tff(fact_9856_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_9857_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),K)) ).

% drop_bit_nat_eq
tff(fact_9858_drop__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9859_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y3: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,Xb),Xaa) = Y3 )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y3 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(nat_prod_decode_aux,aa_nat_nat(suc,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9860_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_9861_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [R3: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S3)))
          & ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),aa_set_nat_nat(finite_card_nat,S3))
             => aa_set_nat_o(member_nat(aa_nat_nat(R3,N5)),S3) ) ) ) ).

% finite_enumerate
tff(fact_9862_fst__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ).

% fst_divmod_nat
tff(fact_9863_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_9864_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_9865_minus__numeral__mod__numeral,axiom,
    ! [Mb: 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,Mb))),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(Mb,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_9866_numeral__mod__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(Mb,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_9867_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(sgn_sgn_rat,P2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(sgn_sgn_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,quotient_of(P2)))),one_one_int) ).

% rat_sgn_code
tff(fact_9868_bezw__non__0,axiom,
    ! [Y3: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y3)
     => ( bezw(Xb,Y3) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y3)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y3)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y3)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y3))))) ) ) ).

% bezw_non_0
tff(fact_9869_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y3: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y3 )
     => ( Y3 = $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_9870_bezw_Osimps,axiom,
    ! [Xb: nat,Y3: nat] :
      bezw(Xb,Y3) = $ite(Y3 = 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(Y3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y3)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y3)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y3)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y3)))))) ).

% bezw.simps
tff(fact_9871_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y3: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y3 )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y3 = $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_9872_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
        $let(
          a2: int,
          a2:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a2)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            a2: int,
            a2:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a2)) ) ) ) ).

% normalize_def
tff(fact_9873_gcd__1__int,axiom,
    ! [Mb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Mb),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9874_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_9875_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_9876_gcd__ge__0__int,axiom,
    ! [Xb: int,Y3: 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),Y3)) ).

% gcd_ge_0_int
tff(fact_9877_bezout__int,axiom,
    ! [Xb: int,Y3: 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),Y3)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y3) ).

% bezout_int
tff(fact_9878_gcd__mult__distrib__int,axiom,
    ! [K: int,Mb: 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,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb)) ).

% gcd_mult_distrib_int
tff(fact_9879_gcd__le2__int,axiom,
    ! [B: int,Aa: 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,Aa),B)),B) ) ).

% gcd_le2_int
tff(fact_9880_gcd__le1__int,axiom,
    ! [Aa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Aa),B)),Aa) ) ).

% gcd_le1_int
tff(fact_9881_gcd__cases__int,axiom,
    ! [Xb: int,Y3: 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),Y3)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y3)) ) )
     => ( ( 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,Y3),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,Y3))) ) )
       => ( ( 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),Y3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),Y3)) ) )
         => ( ( 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,Y3),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,Y3))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y3)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9882_gcd__unique__int,axiom,
    ! [D: int,Aa: 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),Aa)
        & 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),Aa)
              & 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,Aa),B) ) ) ).

% gcd_unique_int
tff(fact_9883_gcd__is__Max__divisors__int,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Mb),Nb) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_xm(Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9884_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun_nat_o,Mb: nat] :
      ( ! [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K3)
         => aa_nat_o(P,K3) )
     => ( ! [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,K3),I4)
                 => aa_nat_o(P,I4) )
             => aa_nat_o(P,K3) ) )
       => aa_nat_o(P,Mb) ) ) ).

% nat_descend_induct
tff(fact_9885_gcd__1__nat,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9886_gcd__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9887_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb))
    <=> ( ( Mb != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9888_gcd__le2__nat,axiom,
    ! [B: nat,Aa: 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,Aa),B)),B) ) ).

% gcd_le2_nat
tff(fact_9889_gcd__le1__nat,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa != 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,Aa),B)),Aa) ) ).

% gcd_le1_nat
tff(fact_9890_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9891_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9892_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Mb: 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,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_9893_bezout__nat,axiom,
    ! [Aa: nat,B: nat] :
      ( ( Aa != zero_zero_nat )
     => ? [X2: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Aa),B)) ) ).

% bezout_nat
tff(fact_9894_bezout__gcd__nat_H,axiom,
    ! [B: nat,Aa: nat] :
    ? [X2: nat,Y: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Aa),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y)),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,Aa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Aa),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9895_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xn(Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9896_bezw__aux,axiom,
    ! [Xb: nat,Y3: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y3)) = 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,Y3))),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,Y3))),aa_nat_int(semiri1314217659103216013at_int,Y3))) ).

% bezw_aux
tff(fact_9897_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y3 )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y3 = $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_9898_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_9899_xor__minus__numerals_I2_J,axiom,
    ! [K: int,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_num_int(neg_numeral_sub_int(Nb),one))) ).

% xor_minus_numerals(2)
tff(fact_9900_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(neg_numeral_sub_int(Nb),one)),K)) ).

% xor_minus_numerals(1)
tff(fact_9901_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9902_sub__BitM__One__eq,axiom,
    ! [Nb: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,Nb)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(Nb),one)) ).

% sub_BitM_One_eq
tff(fact_9903_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or5834768355832116004an_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_9904_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,L),U) = set_or5834768355832116004an_nat(L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9905_tanh__real__bounds,axiom,
    ! [Xb: real] : aa_set_real_o(member_real(aa_real_real(tanh_real,Xb)),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) ).

% tanh_real_bounds
tff(fact_9906_Suc__funpow,axiom,
    ! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_9907_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_xo(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9908_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_di,aTP_Lamm_dj) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9909_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_9910_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_xp) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9911_Inf__real__def,axiom,
    ! [X5: set_real] : comple4887499456419720421f_real(X5) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X5))) ).

% Inf_real_def
tff(fact_9912_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_9913_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_9914_suminf__eq__SUP__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(X5,I2))
       => ( suminf_real(X5) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_xq(X5),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9915_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_9916_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_9917_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_9918_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_xr,Nb),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,Nb) ) ) ).

% range_mod
tff(fact_9919_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9920_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_9921_range__mult,axiom,
    ! [Aa: real] :
      image_real_real(times_times_real(Aa),top_top_set_real) = $ite(Aa = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9922_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_xs,Nb)),Xb)) ).

% root_def
tff(fact_9923_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = 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))))))))) ).

% card_UNIV_char
tff(fact_9924_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_9925_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_9926_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_9927_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))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,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_9928_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = produc4188289175737317920o_char(aTP_Lamm_ya,code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_9929_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_9930_Gcd__eq__Max,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( ( M6 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_yb,M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9931_Gcd__nat__eq__one,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(member_nat(one_one_nat),N3)
     => ( aa_set_nat_nat(gcd_Gcd_nat,N3) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9932_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9933_Gcd__abs__eq,axiom,
    ! [K5: set_int] : gcd_Gcd_int(image_int_int(abs_abs_int,K5)) = gcd_Gcd_int(K5) ).

% Gcd_abs_eq
tff(fact_9934_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_yc,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_9935_Gcd__int__eq,axiom,
    ! [N3: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N3)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N3)) ).

% Gcd_int_eq
tff(fact_9936_DERIV__neg__imp__decreasing,axiom,
    ! [Aa: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,Aa)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9937_DERIV__pos__imp__increasing,axiom,
    ! [Aa: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9938_MVT2,axiom,
    ! [Aa: real,B: real,F: fun_real_real,F5: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F5,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
            & ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,Aa)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),Aa)),aa_real_real(F5,Z3)) ) ) ) ) ).

% MVT2
tff(fact_9939_deriv__nonneg__imp__mono,axiom,
    ! [Aa: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_or1222579329274155063t_real(Aa,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(Aa,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,Aa),B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,Aa)),aa_real_real(G,B)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9940_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [Aa: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,Aa)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9941_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [Aa: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Aa)),aa_real_real(F,B)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9942_DERIV__const__ratio__const,axiom,
    ! [Aa: real,B: real,F: fun_real_real,K: real] :
      ( ( Aa != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,Aa)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),Aa)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9943_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y3: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Xb),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_yd(F),aa_real_real(uminus_uminus_real,Y3),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_9944_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5)) ).

% Gcd_int_greater_eq_0
tff(fact_9945_DERIV__const__average,axiom,
    ! [Aa: real,B: real,V: fun_real_real,K: real] :
      ( ( Aa != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Aa),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(V,Aa)),aa_real_real(V,B))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9946_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Y))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y)),aa_real_real(F,Xb)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9947_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Y))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xb)),aa_real_real(F,Y)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9948_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(divide_divide_real(one_one_real),Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9949_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ye,Nb),aa_real_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,S)) ).

% DERIV_pow
tff(fact_9950_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_yf(G),Nb),aa_real_real(times_times_real(aa_real_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)))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9951_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aTP_Lamm_yg(R2),aa_real_real(times_times_real(R2),aa_real_real(powr_real(Z),aa_real_real(minus_minus_real(R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9952_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_yh(G),R2),aa_real_real(times_times_real(aa_real_real(times_times_real(R2),aa_real_real(powr_real(aa_real_real(G,Xb)),aa_real_real(minus_minus_real(R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9953_DERIV__log,axiom,
    ! [Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(log(B),aa_real_real(divide_divide_real(one_one_real),aa_real_real(times_times_real(aa_real_real(ln_ln_real,B)),Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9954_DERIV__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xb,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_yi(G),F),aa_real_real(times_times_real(aa_real_real(powr_real(aa_real_real(G,Xb)),aa_real_real(F,Xb))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(R2),aa_real_real(ln_ln_real,aa_real_real(G,Xb)))),aa_real_real(divide_divide_real(aa_real_real(times_times_real(Mb),aa_real_real(F,Xb))),aa_real_real(G,Xb)))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9955_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9956_DERIV__arctan,axiom,
    ! [Xb: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9957_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F5: fun_re4238438383274954274t_real,X0: real,Aa: real,B: real,L5: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_mf(F),N),aa_nat_real(aa_real_fun_nat_real(F5,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(Aa,B))
           => summable_real(aa_real_fun_nat_real(F,X2)) )
       => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(Aa,B))
         => ( summable_real(aa_real_fun_nat_real(F5,X0))
           => ( summable_real(L5)
             => ( ! [N: nat,X2: real,Y: real] :
                    ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(Aa,B))
                   => ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(Aa,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(F,X2),N)),aa_nat_real(aa_real_fun_nat_real(F,Y),N)))),aa_real_real(times_times_real(aa_nat_real(L5,N)),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X2),Y)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_sv(F),suminf_real(aa_real_fun_nat_real(F5,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9958_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A)) ).

% arsinh_real_has_field_derivative
tff(fact_9959_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D2: real] :
      ( ( Xb != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( D2 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
           => ( D2 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D2,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9960_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_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))),topolo2177554685111907308n_real(Xb,A)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9961_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_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))))),topolo2177554685111907308n_real(Xb,A)) ) ).

% artanh_real_has_field_derivative
tff(fact_9962_Gcd__int__def,axiom,
    ! [K5: set_int] : gcd_Gcd_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Gcd_int_def
tff(fact_9963_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_yj(F),X2)) )
     => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
         => has_fi5821293074295781190e_real(aTP_Lamm_yl(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_yj(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9964_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9965_DERIV__arccos,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)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_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))))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9966_DERIV__arcsin,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)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_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)))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9967_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M2: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ? [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(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ym(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9968_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ? [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(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ym(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9969_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( Xb != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9970_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T3: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
           => ? [T3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ux(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9971_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ? [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),H)
              & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ux(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9972_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T3: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
           => ? [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),H)
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ux(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9973_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( Xb != zero_zero_real )
         => ( ! [M2: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ? [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(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ym(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9974_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,T3: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
       => ? [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(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ym(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9975_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Aa: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ? [T3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),C)
                  & ( aa_real_real(F,Aa) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yn(Diff),Aa),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Aa),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9976_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Aa: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
             => ? [T3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),B)
                  & ( aa_real_real(F,B) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yn(Diff),B),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(B),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9977_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Aa: real,B: real,C: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),C)
           => ( 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),Xb)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
                 => ( ( Xb != C )
                   => ? [T3: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),C),
                            ( 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),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T3)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),Xb) ) )
                        & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yo(Diff),C),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xb),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9978_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M2: nat,T3: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K) )
       => ! [M: nat,T4: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_yq(Nb),Diff),B2),M),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yr(Diff),M),T4)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M))))),aa_real_real(times_times_real(B2),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,T4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M))))))),topolo2177554685111907308n_real(T4,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9979_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_ys,suminf_real(aa_real_fun_nat_real(aTP_Lamm_yt,Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9980_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xb != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( D2 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
               => ( D2 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
               => ( D2 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D2,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9981_isCont__Lb__Ub,axiom,
    ! [Aa: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => ( ! [X2: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ? [L6: real,M8: real] :
            ( ! [X3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B) )
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_real_real(F,X3))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X3)),M8) ) )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),M8) )
               => ? [X2: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
                    & ( aa_real_real(F,X2) = Y2 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_9982_isCont__real__sqrt,axiom,
    ! [Xb: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_9983_isCont__real__root,axiom,
    ! [Xb: real,Nb: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),aa_nat_fun_real_real(root,Nb)) ).

% isCont_real_root
tff(fact_9984_isCont__inverse__function2,axiom,
    ! [Aa: real,Xb: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
               => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9985_isCont__arcosh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9986_isCont__arccos,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)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9987_isCont__arcsin,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)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9988_LIM__less__bound,axiom,
    ! [B: real,Xb: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Xb)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(B,Xb))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_9989_isCont__artanh,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)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9990_isCont__inverse__function,axiom,
    ! [D: real,Xb: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z3),Xb))),D)
           => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) )
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z3),Xb))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9991_GMVT_H,axiom,
    ! [Aa: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),G) ) )
         => ( ! [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
           => ( ! [Z3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F5,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),B)
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,Aa))),aa_real_real(G2,C3)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B)),aa_real_real(G,Aa))),aa_real_real(F5,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9992_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_9993_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y3: list_int] :
      ( ( upto(Xb,Xaa) = Y3 )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y3 = $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_9994_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_9995_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_9996_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_9997_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9998_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_9999_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_10000_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_10001_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_10002_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_10003_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_10004_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_10005_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_10006_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_10007_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_10008_distinct__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(distinct_int,upto(I,J)) ).

% distinct_upto
tff(fact_10009_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_10010_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_10011_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_10012_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_10013_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_10014_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y3: list_int] :
      ( ( upto(Xb,Xaa) = Y3 )
     => ( Y3 = $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_10015_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_10016_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_10017_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_10018_GMVT,axiom,
    ! [Aa: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( ! [X2: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ( ! [X2: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) )
         => ( ! [X2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),G) )
           => ( ! [X2: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X2,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),B)
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,Aa))),G_c) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B)),aa_real_real(G,Aa))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_10019_MVT,axiom,
    ! [Aa: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,B),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ? [L3: real,Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z3,top_top_set_real))
              & ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,Aa)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),Aa)),L3) ) ) ) ) ) ).

% MVT
tff(fact_10020_continuous__on__arsinh_H,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A,F)
     => topolo5044208981011980120l_real(A,aTP_Lamm_yu(F)) ) ).

% continuous_on_arsinh'
tff(fact_10021_continuous__on__arcosh_H,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A,F)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X2)) )
       => topolo5044208981011980120l_real(A,aTP_Lamm_yv(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_10022_continuous__image__closed__interval,axiom,
    ! [Aa: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,B),F)
       => ? [C3: real,D3: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(Aa,B)) = set_or1222579329274155063t_real(C3,D3) )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C3),D3) ) ) ) ).

% continuous_image_closed_interval
tff(fact_10023_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arccos) ).

% continuous_on_arccos'
tff(fact_10024_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arcsin) ).

% continuous_on_arcsin'
tff(fact_10025_continuous__on__artanh,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A,artanh_real) ) ).

% continuous_on_artanh
tff(fact_10026_continuous__on__artanh_H,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A,F)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A)
           => aa_set_real_o(member_real(aa_real_real(F,X2)),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) )
       => topolo5044208981011980120l_real(A,aTP_Lamm_yw(F)) ) ) ).

% continuous_on_artanh'
tff(fact_10027_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_yx,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_10028_DERIV__isconst2,axiom,
    ! [Aa: real,B: real,F: fun_real_real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,B),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Xb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
             => ( aa_real_real(F,Xb) = aa_real_real(F,Aa) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_10029_summable__Leibniz_I3_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(Aa,zero_zero_nat)),zero_zero_real)
         => ! [N5: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_yy(Aa))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_10030_summable__Leibniz_I2_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(Aa,zero_zero_nat))
         => ! [N5: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_yy(Aa))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_10031_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_10032_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_10033_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ao,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_10034_monoseq__convergent,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( topolo6980174941875973593q_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X5,I2))),B2)
       => ~ ! [L6: real] : ~ filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_10035_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_yz,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_10036_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kn(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N5)),L3)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N5))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_10037_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun_nat_real] :
      ( ! [R3: real] :
        ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X5,N)) )
     => filterlim_nat_real(aTP_Lamm_za(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_10038_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_zb,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_10039_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zc,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_10040_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_zd,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_10041_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ze,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_10042_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
       => ( ! [E2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
             => ? [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(plus_plus_real(aa_nat_real(F,N5)),E2)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_10043_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_10044_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zf(Xb),Aa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_10045_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_10046_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_10047_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zg,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_10048_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zh,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_10049_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zi,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_10050_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zj,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_10051_summable__Leibniz_I1_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => summable_real(aTP_Lamm_yy(Aa)) ) ) ).

% summable_Leibniz(1)
tff(fact_10052_summable,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => summable_real(aTP_Lamm_yy(Aa)) ) ) ) ).

% summable
tff(fact_10053_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zk(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K3: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_zl(Theta),K3),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_10054_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_zm(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K3: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_zl(Theta),K3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_10055_summable__Leibniz_I4_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => filterlim_nat_real(aTP_Lamm_zn(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_yy(Aa))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_10056_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_qg,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_10057_summable__Leibniz_H_I2_J,axiom,
    ! [Aa: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),suminf_real(aTP_Lamm_yy(Aa))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_10058_summable__Leibniz_H_I3_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => filterlim_nat_real(aTP_Lamm_zn(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_yy(Aa))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_10059_sums__alternating__upper__lower,axiom,
    ! [Aa: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)))),L3)
              & filterlim_nat_real(aTP_Lamm_zn(Aa),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_zo(Aa),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_10060_summable__Leibniz_I5_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => filterlim_nat_real(aTP_Lamm_zo(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_yy(Aa))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_10061_summable__Leibniz_H_I4_J,axiom,
    ! [Aa: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_yy(Aa))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_10062_summable__Leibniz_H_I5_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => filterlim_nat_real(aTP_Lamm_zo(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_yy(Aa))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_10063_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C4: real] :
        ! [X: real] : aa_real_real(F,X) = aa_real_real(times_times_real(X),C4) ) ).

% real_bounded_linear
tff(fact_10064_tendsto__exp__limit__at__right,axiom,
    ! [Xb: real] : filterlim_real_real(aTP_Lamm_zp(Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_10065_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real(arcosh_real,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(one_one_real,set_or5849166863359141190n_real(one_one_real))) ).

% tendsto_arcosh_at_left_1
tff(fact_10066_greaterThan__0,axiom,
    aa_nat_set_nat(set_or1210151606488870762an_nat,zero_zero_nat) = image_nat_nat(suc,top_top_set_nat) ).

% greaterThan_0
tff(fact_10067_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_10068_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_10069_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_10070_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_10071_tanh__real__at__bot,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,one_one_real)),at_bot_real) ).

% tanh_real_at_bot
tff(fact_10072_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_10073_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_10074_artanh__real__at__right__1,axiom,
    filterlim_real_real(artanh_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,one_one_real),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,one_one_real)))) ).

% artanh_real_at_right_1
tff(fact_10075_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
         => ? [Y2: real] :
              ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_10076_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zq(Nb),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_10077_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zq(Nb),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_10078_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_10079_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_10080_ln__at__top,axiom,
    filterlim_real_real(ln_ln_real,at_top_real,at_top_real) ).

% ln_at_top
tff(fact_10081_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_10082_exp__at__top,axiom,
    filterlim_real_real(exp_real,at_top_real,at_top_real) ).

% exp_at_top
tff(fact_10083_filterlim__real__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_top_real,at_top_nat) ).

% filterlim_real_sequentially
tff(fact_10084_filterlim__uminus__at__top__at__bot,axiom,
    filterlim_real_real(uminus_uminus_real,at_top_real,at_bot_real) ).

% filterlim_uminus_at_top_at_bot
tff(fact_10085_filterlim__uminus__at__bot__at__top,axiom,
    filterlim_real_real(uminus_uminus_real,at_bot_real,at_top_real) ).

% filterlim_uminus_at_bot_at_top
tff(fact_10086_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_10087_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_zr,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_10088_artanh__real__at__left__1,axiom,
    filterlim_real_real(artanh_real,at_top_real,topolo2177554685111907308n_real(one_one_real,set_or5984915006950818249n_real(one_one_real))) ).

% artanh_real_at_left_1
tff(fact_10089_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_10090_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_10091_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_zs,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_10092_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim_real_real(aTP_Lamm_zt(Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10093_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10094_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X2)
         => ? [Y2: real] :
              ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_10095_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10096_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y3: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( eventually_real(aTP_Lamm_zu(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G2),F5),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G),F),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_10097_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_zx(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_10098_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_zy(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_10099_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_zy(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_10100_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_az,at_top_nat) ).

% eventually_False_sequentially
tff(fact_10101_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F3) ) ).

% le_sequentially
tff(fact_10102_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N8: nat] :
        ! [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N4)
         => aa_nat_o(P,N4) ) ) ).

% eventually_sequentially
tff(fact_10103_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X2)
         => aa_nat_o(P,X2) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_10104_eventually__at__right__to__0,axiom,
    ! [P: fun_real_o,Aa: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
    <=> eventually_real(aa_real_fun_real_o(aTP_Lamm_zz(P),Aa),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_10105_eventually__at__left__to__right,axiom,
    ! [P: fun_real_o,Aa: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
    <=> eventually_real(aTP_Lamm_aaa(P),topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Aa),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,Aa)))) ) ).

% eventually_at_left_to_right
tff(fact_10106_eventually__at__right__real,axiom,
    ! [Aa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_aab(Aa),B),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa))) ) ).

% eventually_at_right_real
tff(fact_10107_eventually__at__left__real,axiom,
    ! [B: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Aa)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_aab(B),Aa),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa))) ) ).

% eventually_at_left_real
tff(fact_10108_eventually__at__top__to__right,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_aac(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_10109_eventually__at__right__to__top,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_aac(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_10110_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Aa,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Aa,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F5),G2),at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F),G),at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_10111_lhopital,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aTP_Lamm_zu(G),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aTP_Lamm_zu(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G2),F5),F3,topolo2177554685111907308n_real(Xb,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F),G),F3,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_10112_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F5),G2),at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F),G),at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_10113_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Aa,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Aa,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F5),G2),at_bot_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F),G),at_bot_real,topolo2177554685111907308n_real(Aa,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_10114_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F5),G2),at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F),G),at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_10115_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Xb: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_zu(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G2),F5),topolo2815343760600316023s_real(Xb),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G),F),topolo2815343760600316023s_real(Xb),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_10116_lhopital__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y3: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( eventually_real(aTP_Lamm_zu(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G2),F5),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G),F),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_10117_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_zu(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_zu(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F0),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G2),F5),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F0),G0),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_10118_lhopital__right,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
       => ( eventually_real(aTP_Lamm_zu(G),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
         => ( eventually_real(aTP_Lamm_zu(G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G2),F5),F3,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F),G),F3,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_10119_lhopital__left,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aTP_Lamm_zu(G),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aTP_Lamm_zu(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G2),F5),F3,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F),G),F3,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_10120_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F5),G2),at_bot_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F),G),at_bot_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_10121_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F5),G2),at_bot_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(F),G),at_bot_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_10122_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Xb: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_zu(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G2),F5),topolo2815343760600316023s_real(Xb),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G),F),topolo2815343760600316023s_real(Xb),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_10123_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y3: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
     => ( eventually_real(aTP_Lamm_zu(G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(F),F5),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_zv(G),G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G2),F5),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(G),F),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_10124_Bseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xb),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_10125_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,Aa: real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(image_nat_real(F,top_top_set_nat)),set_or1222579329274155063t_real(Aa,B))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_10126_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_10127_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_10128_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_10129_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,L),U) = set_or6659071591806873216st_nat(L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_10130_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_10131_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_10132_decseq__bounded,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_nat_real(X5,I2))
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_10133_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_10134_decseq__convergent,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_nat_real(X5,I2))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_nat_real(X5,I4)) ) ) ) ).

% decseq_convergent
tff(fact_10135_UN__atLeast__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_10136_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_10137_continuous__on__arcosh,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_10138_take__bit__numeral__minus__numeral__int,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_aae(Mb),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Mb),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_10139_take__bit__num__simps_I1_J,axiom,
    ! [Mb: num] : bit_take_bit_num(zero_zero_nat,Mb) = none_num ).

% take_bit_num_simps(1)
tff(fact_10140_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_10141_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_10142_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_aaf,bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_10143_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_10144_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_aaf,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb)) ).

% take_bit_num_simps(6)
tff(fact_10145_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb))) ).

% take_bit_num_simps(7)
tff(fact_10146_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Mb)) = case_nat_option_num(none_num,aTP_Lamm_aag(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_10147_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one) = case_nat_option_num(none_num,aTP_Lamm_aah,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_10148_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Mb)) = case_nat_option_num(none_num,aTP_Lamm_aai(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_10149_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(7)
tff(fact_10150_and__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(3)
tff(fact_10151_and__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_10152_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_10153_open__bool__def,axiom,
    topolo9180104560040979295open_o = topolo4667128019001906403logy_o(sup_sup_set_set_o(image_o_set_o(set_ord_lessThan_o,top_top_set_o),image_o_set_o(set_or6416164934427428222Than_o,top_top_set_o))) ).

% open_bool_def
tff(fact_10154_open__int__def,axiom,
    topolo4325760605701065253en_int = topolo1611008123915946401gy_int(sup_sup_set_set_int(image_int_set_int(set_ord_lessThan_int,top_top_set_int),image_int_set_int(set_or1207661135979820486an_int,top_top_set_int))) ).

% open_int_def
tff(fact_10155_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( set_or4665077453230672383an_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(set_or4665077453230672383an_nat(I,J),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_10156_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_10157_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_10158_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_10159_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_10160_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit1,Nb)) = none_num ).

% and_not_num.simps(3)
tff(fact_10161_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit0,Nb)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_10162_and__not__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit0,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(4)
tff(fact_10163_and__not__num_Osimps_I7_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit1,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(7)
tff(fact_10164_and__not__num__eq__Some__iff,axiom,
    ! [Mb: num,Nb: num,Q2: num] :
      ( ( bit_and_not_num(Mb,Nb) = some_num(Q2) )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_10165_and__not__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aaj,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(8)
tff(fact_10166_and__not__num__eq__None__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( bit_and_not_num(Mb,Nb) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_10167_open__nat__def,axiom,
    topolo4328251076210115529en_nat = topolo1613498594424996677gy_nat(sup_sup_set_set_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat),image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat))) ).

% open_nat_def
tff(fact_10168_int__numeral__not__and__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,Mb)) ).

% int_numeral_not_and_num
tff(fact_10169_int__numeral__and__not__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,Nb)) ).

% int_numeral_and_not_num
tff(fact_10170_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,Mb) = produc478579273971653890on_num(aTP_Lamm_aan,product_Pair_nat_num(Nb,Mb)) ).

% Bit_Operations.take_bit_num_code
tff(fact_10171_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y3) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y3)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y3)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X),n) )
                          & aa_nat_o(vEBT_VEBT_valid(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) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_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(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_aao(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_10172_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg3: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m2: nat,
              m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Dega),n),
              ( ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X),n) )
              & aa_nat_o(vEBT_VEBT_valid(Summarya),m2)
              & ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_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(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_aao(Dega),TreeLista),Summarya),n),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_10173_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X),n) )
                      & aa_nat_o(vEBT_VEBT_valid(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) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_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(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_aao(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_10174_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X),n) )
                        & aa_nat_o(vEBT_VEBT_valid(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) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_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(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_aao(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_10175_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X),n) )
                          & aa_nat_o(vEBT_VEBT_valid(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) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_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(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_aao(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_10176_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X),n) )
                            & aa_nat_o(vEBT_VEBT_valid(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) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_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(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_aao(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_10177_Sup__int__def,axiom,
    ! [X5: set_int] : complete_Sup_Sup_int(X5) = the_int(aTP_Lamm_aap(X5)) ).

% Sup_int_def
tff(fact_10178_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y3: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y3) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y3)
                <=> ( Xaa = one_one_nat ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y3)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X),n) )
                            & aa_nat_o(vEBT_VEBT_valid(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) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_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(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_aao(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_10179_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb))))) ).

% take_bit_num_def
tff(fact_10180_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_10181_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_10182_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_10183_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_10184_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_10185_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,Mb)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_10186_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_10187_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_aar,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_10188_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_aat,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_10189_open__complex__def,axiom,
    ! [U3: set_complex] :
      ( aa_set_complex_o(topolo4110288021797289639omplex,U3)
    <=> ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),U3)
         => eventu5826381225784669381omplex(produc6771430404735790350plex_o(aa_com7221863818706917250plex_o(aTP_Lamm_aau(U3),X)),topolo896644834953643431omplex) ) ) ).

% open_complex_def
tff(fact_10190_open__real__def,axiom,
    ! [U3: set_real] :
      ( aa_set_real_o(topolo4860482606490270245n_real,U3)
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),U3)
         => eventu3244425730907250241l_real(produc5414030515140494994real_o(aa_rea1652550732481586312real_o(aTP_Lamm_aav(U3),X)),topolo1511823702728130853y_real) ) ) ).

% open_real_def
tff(fact_10191_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N8: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M5)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N4)
             => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,N4),M5)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_10192_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).

% mono_times_nat
tff(fact_10193_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_10194_incseq__bounded,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I2)),B2)
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_10195_incseq__convergent,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I2)),B2)
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I4)),L6) ) ) ) ).

% incseq_convergent
tff(fact_10196_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aaw,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_10197_tendsto__at__topI__sequentially__real,axiom,
    ! [F: fun_real_real,Y3: real] :
      ( order_mono_real_real(F)
     => ( filterlim_nat_real(aTP_Lamm_aax(F),topolo2815343760600316023s_real(Y3),at_top_nat)
       => filterlim_real_real(F,topolo2815343760600316023s_real(Y3),at_top_real) ) ) ).

% tendsto_at_topI_sequentially_real
tff(fact_10198_filtermap__at__right__shift,axiom,
    ! [D: real,Aa: real] : filtermap_real_real(aTP_Lamm_aay(D),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa))) = topolo2177554685111907308n_real(aa_real_real(minus_minus_real(Aa),D),set_or5849166863359141190n_real(aa_real_real(minus_minus_real(Aa),D))) ).

% filtermap_at_right_shift
tff(fact_10199_at__right__to__0,axiom,
    ! [Aa: real] : topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)) = filtermap_real_real(aTP_Lamm_aaz(Aa),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% at_right_to_0
tff(fact_10200_at__left__minus,axiom,
    ! [Aa: real] : topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)) = filtermap_real_real(uminus_uminus_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Aa),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,Aa)))) ).

% at_left_minus
tff(fact_10201_at__right__minus,axiom,
    ! [Aa: real] : topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)) = filtermap_real_real(uminus_uminus_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Aa),set_or5984915006950818249n_real(aa_real_real(uminus_uminus_real,Aa)))) ).

% at_right_minus
tff(fact_10202_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_qo,Uu) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_10203_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_vf,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10204_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_vz,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10205_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ur,Uu) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_10206_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_yx,Uu) = aa_real_real(divide_divide_real(aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10207_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_zr,Uu) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_10208_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_yz,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_7
tff(fact_10209_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_zb,Uu) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_8
tff(fact_10210_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ys,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_qi,Uu)) ).

% ATP.lambda_9
tff(fact_10211_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_yb,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gd,Uu)) ).

% ATP.lambda_10
tff(fact_10212_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_aat,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_aas(Uu)))) ).

% ATP.lambda_11
tff(fact_10213_ATP_Olambda__12,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_aar,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_aaq(Uu)))) ).

% ATP.lambda_12
tff(fact_10214_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_zd,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_13
tff(fact_10215_ATP_Olambda__14,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_aaj,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_14
tff(fact_10216_ATP_Olambda__15,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_aaf,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_15
tff(fact_10217_ATP_Olambda__16,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_wh,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_16
tff(fact_10218_ATP_Olambda__17,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_wg,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_17
tff(fact_10219_ATP_Olambda__18,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_he,Uu) = aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(suc,Uu)) ).

% ATP.lambda_18
tff(fact_10220_ATP_Olambda__19,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_yc,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_19
tff(fact_10221_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_aah,Uu) = some_num(one) ).

% ATP.lambda_20
tff(fact_10222_ATP_Olambda__21,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_aam(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_aak(Uua),aTP_Lamm_aal(Uua),Uu) ).

% ATP.lambda_21
tff(fact_10223_ATP_Olambda__22,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_vg(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_22
tff(fact_10224_ATP_Olambda__23,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_we,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_23
tff(fact_10225_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_aan,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_aam(Uua),Uu) ).

% ATP.lambda_24
tff(fact_10226_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_aak(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_aaf,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_25
tff(fact_10227_ATP_Olambda__26,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_aag(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_aaf,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_26
tff(fact_10228_ATP_Olambda__27,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qh,Uu),Uua) = aa_real_real(times_times_real(aa_real_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(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(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_27
tff(fact_10229_ATP_Olambda__28,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ve,Uu),Uua) = aa_real_real(times_times_real(aa_real_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_28
tff(fact_10230_ATP_Olambda__29,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vi,Uu),Uua) = aa_real_real(times_times_real(aa_real_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_29
tff(fact_10231_ATP_Olambda__30,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vw,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_30
tff(fact_10232_ATP_Olambda__31,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vv,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_31
tff(fact_10233_ATP_Olambda__32,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_xa,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_32
tff(fact_10234_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_qf,Uu),Uua) = $let(
        m3: code_integer,
        m3:= 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,m3,plus_p5714425477246183910nteger(m3,one_one_Code_integer)) ) ).

% ATP.lambda_33
tff(fact_10235_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qa,Uu),Uua) = $let(
        m3: complex,
        m3:= aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,m3),one_one_complex)) ) ).

% ATP.lambda_34
tff(fact_10236_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qd,Uu),Uua) = $let(
        m3: real,
        m3:= aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_real_real(plus_plus_real(m3),one_one_real)) ) ).

% ATP.lambda_35
tff(fact_10237_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_qb,Uu),Uua) = $let(
        m3: rat,
        m3:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_rat_rat(plus_plus_rat(m3),one_one_rat)) ) ).

% ATP.lambda_36
tff(fact_10238_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qe,Uu),Uua) = $let(
        m3: nat,
        m3:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m3),one_one_nat)) ) ).

% ATP.lambda_37
tff(fact_10239_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qc,Uu),Uua) = $let(
        m3: int,
        m3:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_int_int(aa_int_fun_int_int(plus_plus_int,m3),one_one_int)) ) ).

% ATP.lambda_38
tff(fact_10240_ATP_Olambda__39,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_wd(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_39
tff(fact_10241_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aa_real_fun_int_o(aTP_Lamm_wa,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_40
tff(fact_10242_ATP_Olambda__41,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_wc(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_41
tff(fact_10243_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qi,Uu),Uua) = aa_real_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(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(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_42
tff(fact_10244_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yt,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_43
tff(fact_10245_ATP_Olambda__44,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yy(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_44
tff(fact_10246_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_xp,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_45
tff(fact_10247_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vs,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_46
tff(fact_10248_ATP_Olambda__47,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_aap(Uu),Uua)
    <=> ( aa_set_int_o(member_int(Uua),Uu)
        & ! [X: int] :
            ( aa_set_int_o(member_int(X),Uu)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Uua) ) ) ) ).

% ATP.lambda_47
tff(fact_10249_ATP_Olambda__48,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vd,Uu),Uua) = aa_real_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_48
tff(fact_10250_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_zs,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_49
tff(fact_10251_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aaw,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_50
tff(fact_10252_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vl,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_51
tff(fact_10253_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vk,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_52
tff(fact_10254_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_do,Uu),Uua)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_53
tff(fact_10255_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_dn(Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_54
tff(fact_10256_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zi,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_55
tff(fact_10257_ATP_Olambda__56,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_zp(Uu),Uua) = aa_real_real(powr_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(Uu),Uua))),aa_real_real(divide_divide_real(one_one_real),Uua)) ).

% ATP.lambda_56
tff(fact_10258_ATP_Olambda__57,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_zt(Uu),Uua) = aa_real_real(powr_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Uu),Uua))),Uua) ).

% ATP.lambda_57
tff(fact_10259_ATP_Olambda__58,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qg,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(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_10260_ATP_Olambda__59,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vy,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_59
tff(fact_10261_ATP_Olambda__60,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ph(Uu),Uua) = aa_real_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_60
tff(fact_10262_ATP_Olambda__61,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pe(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_61
tff(fact_10263_ATP_Olambda__62,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_pg(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_62
tff(fact_10264_ATP_Olambda__63,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pf(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_63
tff(fact_10265_ATP_Olambda__64,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ow(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_64
tff(fact_10266_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ou(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_65
tff(fact_10267_ATP_Olambda__66,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ov(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_66
tff(fact_10268_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_nx(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Uua)) ).

% ATP.lambda_67
tff(fact_10269_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nz(Uu),Uua) = aa_real_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_68
tff(fact_10270_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ny(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_69
tff(fact_10271_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_te(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),Uua)) ).

% ATP.lambda_70
tff(fact_10272_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pd(Uu),Uua) = aa_real_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_71
tff(fact_10273_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pb(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_72
tff(fact_10274_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pc(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_73
tff(fact_10275_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ox(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(Uu,Uua)),aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_74
tff(fact_10276_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pa(Uu),Uua) = aa_real_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_75
tff(fact_10277_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oy(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_76
tff(fact_10278_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_oz(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_77
tff(fact_10279_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_to(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_78
tff(fact_10280_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_tm(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_79
tff(fact_10281_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_tn(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_80
tff(fact_10282_ATP_Olambda__81,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_mv(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_81
tff(fact_10283_ATP_Olambda__82,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_mx(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_82
tff(fact_10284_ATP_Olambda__83,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_mz(Uu),Uua)
    <=> ( aa_complex_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_83
tff(fact_10285_ATP_Olambda__84,axiom,
    ! [Uu: fun_complex_int,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_na(Uu),Uua)
    <=> ( aa_complex_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_84
tff(fact_10286_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_nb(Uu),Uua)
    <=> ( aa_nat_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_85
tff(fact_10287_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_nc(Uu),Uua)
    <=> ( aa_nat_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_86
tff(fact_10288_ATP_Olambda__87,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( aa_int_o(aTP_Lamm_mt(Uu),Uua)
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_87
tff(fact_10289_ATP_Olambda__88,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_mu(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_88
tff(fact_10290_ATP_Olambda__89,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_mw(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_89
tff(fact_10291_ATP_Olambda__90,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_my(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_90
tff(fact_10292_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zo(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_91
tff(fact_10293_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zn(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yy(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_92
tff(fact_10294_ATP_Olambda__93,axiom,
    ! [Uu: code_integer,Uua: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_ya,Uu),(Uua)) = produc4188289175737317920o_char(aTP_Lamm_xz((Uua)),code_bit_cut_integer(Uu)) ).

% ATP.lambda_93
tff(fact_10295_ATP_Olambda__94,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_aae(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_94
tff(fact_10296_ATP_Olambda__95,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_xs,Uu),Uua) = aa_real_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_95
tff(fact_10297_ATP_Olambda__96,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vc,Uu),Uua) = aa_real_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_96
tff(fact_10298_ATP_Olambda__97,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uv,Uu),Uua) = aa_real_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_97
tff(fact_10299_ATP_Olambda__98,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_xb,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa_num_num(aa_num_fun_num_num(plus_plus_num,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(aa_num_fun_num_num(plus_plus_num,l3),one)) ) ) ).

% ATP.lambda_98
tff(fact_10300_ATP_Olambda__99,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_xd,Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_99
tff(fact_10301_ATP_Olambda__100,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vx,Uu),Uua)
    <=> ( aa_real_real(exp_real,Uua) = Uu ) ) ).

% ATP.lambda_100
tff(fact_10302_ATP_Olambda__101,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gi,Uu),Uua) = produc1086072967326762835nteger(Uu,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_101
tff(fact_10303_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gg,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_102
tff(fact_10304_ATP_Olambda__103,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gh,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_103
tff(fact_10305_ATP_Olambda__104,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gl,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_104
tff(fact_10306_ATP_Olambda__105,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gj,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_105
tff(fact_10307_ATP_Olambda__106,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gk,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_106
tff(fact_10308_ATP_Olambda__107,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vt,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_107
tff(fact_10309_ATP_Olambda__108,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zj,Uu),Uua) = aa_real_real(times_times_real(Uu),aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_108
tff(fact_10310_ATP_Olambda__109,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_hv,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_109
tff(fact_10311_ATP_Olambda__110,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zh,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_110
tff(fact_10312_ATP_Olambda__111,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ze,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_111
tff(fact_10313_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xq(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(Uu),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_112
tff(fact_10314_ATP_Olambda__113,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_dl(Uu),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu) ) ).

% ATP.lambda_113
tff(fact_10315_ATP_Olambda__114,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_dk(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_114
tff(fact_10316_ATP_Olambda__115,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_dm(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_115
tff(fact_10317_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_di,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_116
tff(fact_10318_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xr,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_117
tff(fact_10319_ATP_Olambda__118,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_so,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_118
tff(fact_10320_ATP_Olambda__119,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_sp((Uu)),(Uua))
    <=> aa_o_o(ord_less_o((Uua)),(Uu)) ) ).

% ATP.lambda_119
tff(fact_10321_ATP_Olambda__120,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_sk(Uu),Uua)
    <=> aa_set_nat_o(ord_less_set_nat(Uua),Uu) ) ).

% ATP.lambda_120
tff(fact_10322_ATP_Olambda__121,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_sl,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_121
tff(fact_10323_ATP_Olambda__122,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_sm,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu) ) ).

% ATP.lambda_122
tff(fact_10324_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dj,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_123
tff(fact_10325_ATP_Olambda__124,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_sn,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_124
tff(fact_10326_ATP_Olambda__125,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_am(Uu),Uua) = aa_real_real(times_times_real(Uua),Uu) ).

% ATP.lambda_125
tff(fact_10327_ATP_Olambda__126,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_an(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_126
tff(fact_10328_ATP_Olambda__127,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ao,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_127
tff(fact_10329_ATP_Olambda__128,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ap,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).

% ATP.lambda_128
tff(fact_10330_ATP_Olambda__129,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_aay(Uu),Uua) = aa_real_real(minus_minus_real(Uua),Uu) ).

% ATP.lambda_129
tff(fact_10331_ATP_Olambda__130,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wu,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_130
tff(fact_10332_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ye,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_131
tff(fact_10333_ATP_Olambda__132,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_aaz(Uu),Uua) = aa_real_real(plus_plus_real(Uua),Uu) ).

% ATP.lambda_132
tff(fact_10334_ATP_Olambda__133,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ww,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_133
tff(fact_10335_ATP_Olambda__134,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_yg(Uu),Uua) = aa_real_real(powr_real(Uua),Uu) ).

% ATP.lambda_134
tff(fact_10336_ATP_Olambda__135,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_ge,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_135
tff(fact_10337_ATP_Olambda__136,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_gc,Uu),Uua)
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Uua),Uu) ) ).

% ATP.lambda_136
tff(fact_10338_ATP_Olambda__137,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gd,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_137
tff(fact_10339_ATP_Olambda__138,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_gf,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_138
tff(fact_10340_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_139
tff(fact_10341_ATP_Olambda__140,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zc,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_140
tff(fact_10342_ATP_Olambda__141,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aa(Uu),Uua)
    <=> member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_141
tff(fact_10343_ATP_Olambda__142,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ab(Uu),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_142
tff(fact_10344_ATP_Olambda__143,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_143
tff(fact_10345_ATP_Olambda__144,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_10346_ATP_Olambda__145,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_145
tff(fact_10347_ATP_Olambda__146,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_146
tff(fact_10348_ATP_Olambda__147,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_bi(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_147
tff(fact_10349_ATP_Olambda__148,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aTP_Lamm_bj,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_148
tff(fact_10350_ATP_Olambda__149,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_bk,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_149
tff(fact_10351_ATP_Olambda__150,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bo,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_150
tff(fact_10352_ATP_Olambda__151,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_bl(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_151
tff(fact_10353_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bm,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_152
tff(fact_10354_ATP_Olambda__153,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bn,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_153
tff(fact_10355_ATP_Olambda__154,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uq(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_154
tff(fact_10356_ATP_Olambda__155,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_up(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_155
tff(fact_10357_ATP_Olambda__156,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_aac(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(inverse_inverse_real,Uua)) ) ).

% ATP.lambda_156
tff(fact_10358_ATP_Olambda__157,axiom,
    ! [Uu: fun_real_real,Uua: nat] : aa_nat_real(aTP_Lamm_aax(Uu),Uua) = aa_real_real(Uu,aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_157
tff(fact_10359_ATP_Olambda__158,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_yd(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_158
tff(fact_10360_ATP_Olambda__159,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_aaa(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(uminus_uminus_real,Uua)) ) ).

% ATP.lambda_159
tff(fact_10361_ATP_Olambda__160,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rq(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_160
tff(fact_10362_ATP_Olambda__161,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oj(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_161
tff(fact_10363_ATP_Olambda__162,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zx(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_162
tff(fact_10364_ATP_Olambda__163,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_os(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_163
tff(fact_10365_ATP_Olambda__164,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_oi(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_164
tff(fact_10366_ATP_Olambda__165,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ot(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_165
tff(fact_10367_ATP_Olambda__166,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_aal(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_166
tff(fact_10368_ATP_Olambda__167,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_aai(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_167
tff(fact_10369_ATP_Olambda__168,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_yl(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_yk(Uu),Uua)) ).

% ATP.lambda_168
tff(fact_10370_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_vu,Uu),Uua) = aa_real_complex(cis,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_169
tff(fact_10371_ATP_Olambda__170,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_zu(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_170
tff(fact_10372_ATP_Olambda__171,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hf,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_171
tff(fact_10373_ATP_Olambda__172,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zg,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_172
tff(fact_10374_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xk,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_173
tff(fact_10375_ATP_Olambda__174,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xl,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_174
tff(fact_10376_ATP_Olambda__175,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_hl(Uu),Uua)
    <=> ~ member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_175
tff(fact_10377_ATP_Olambda__176,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_hm(Uu),Uua)
    <=> ~ aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_176
tff(fact_10378_ATP_Olambda__177,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_hk(Uu),Uua)
    <=> ~ aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_177
tff(fact_10379_ATP_Olambda__178,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_hn(Uu),Uua)
    <=> ~ aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_178
tff(fact_10380_ATP_Olambda__179,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ho(Uu),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_179
tff(fact_10381_ATP_Olambda__180,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_hp(Uu),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_10382_ATP_Olambda__181,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_qk(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_181
tff(fact_10383_ATP_Olambda__182,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_qj(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_182
tff(fact_10384_ATP_Olambda__183,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ql(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_183
tff(fact_10385_ATP_Olambda__184,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_za(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_184
tff(fact_10386_ATP_Olambda__185,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_pp(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_185
tff(fact_10387_ATP_Olambda__186,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_pr(Uu),Uua) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_186
tff(fact_10388_ATP_Olambda__187,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_pt(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_187
tff(fact_10389_ATP_Olambda__188,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ps(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_188
tff(fact_10390_ATP_Olambda__189,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_pq(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_189
tff(fact_10391_ATP_Olambda__190,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_po(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_190
tff(fact_10392_ATP_Olambda__191,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_complex(aTP_Lamm_ut(Uu),Uua) = aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_191
tff(fact_10393_ATP_Olambda__192,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_us(Uu),Uua) = aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_192
tff(fact_10394_ATP_Olambda__193,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uu(Uu),Uua) = aa_real_real(real_V1803761363581548252l_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_193
tff(fact_10395_ATP_Olambda__194,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_complex(aTP_Lamm_kr(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_194
tff(fact_10396_ATP_Olambda__195,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rs(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_195
tff(fact_10397_ATP_Olambda__196,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ks(Uu),Uua) = aa_real_real(uminus_uminus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_196
tff(fact_10398_ATP_Olambda__197,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_kq(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_197
tff(fact_10399_ATP_Olambda__198,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_ja(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_198
tff(fact_10400_ATP_Olambda__199,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_jb(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_199
tff(fact_10401_ATP_Olambda__200,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_jc(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_200
tff(fact_10402_ATP_Olambda__201,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_jd(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_201
tff(fact_10403_ATP_Olambda__202,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_je(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_202
tff(fact_10404_ATP_Olambda__203,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_pj(Uu),Uua) = aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_203
tff(fact_10405_ATP_Olambda__204,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_pn(Uu),Uua) = plus_plus_real(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_204
tff(fact_10406_ATP_Olambda__205,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_pk(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_205
tff(fact_10407_ATP_Olambda__206,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_pm(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_206
tff(fact_10408_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_pl(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_207
tff(fact_10409_ATP_Olambda__208,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_yw(Uu),Uua) = aa_real_real(artanh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_208
tff(fact_10410_ATP_Olambda__209,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_yu(Uu),Uua) = aa_real_real(arsinh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_209
tff(fact_10411_ATP_Olambda__210,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_yv(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_10412_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_if(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_10413_ATP_Olambda__212,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ie(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_10414_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zm(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_10415_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_wx(Uu),Uua) = aa_complex_real(re,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_10416_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_wy(Uu),Uua) = aa_complex_real(im,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_10417_ATP_Olambda__216,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: complex] : aa_complex_complex(aTP_Lamm_tb(Uu),Uua) = suminf_complex(aa_com8181001469867462503omplex(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_10418_ATP_Olambda__217,axiom,
    ! [Uu: fun_co8378349014452122532t_real,Uua: complex] : aa_complex_real(aTP_Lamm_su(Uu),Uua) = suminf_real(aa_com20334259897932645t_real(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_10419_ATP_Olambda__218,axiom,
    ! [Uu: fun_re2084383175013312036omplex,Uua: real] : aa_real_complex(aTP_Lamm_sx(Uu),Uua) = suminf_complex(aa_rea1531280491672885477omplex(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_10420_ATP_Olambda__219,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_sv(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_10421_ATP_Olambda__220,axiom,
    ! [Uu: fun_na1142814060743658952omplex,Uua: nat] : aa_nat_complex(aTP_Lamm_sy(Uu),Uua) = suminf_complex(aa_nat6454746962644518025omplex(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_10422_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_td(Uu),Uua) = suminf_real(aa_nat_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_10423_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_tc(Uu),Uua) = suminf_nat(aa_nat_fun_nat_nat(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_10424_ATP_Olambda__223,axiom,
    ! [Uu: fun_in8056368165505483812omplex,Uua: int] : aa_int_complex(aTP_Lamm_sz(Uu),Uua) = suminf_complex(aa_int4144929030551567077omplex(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_10425_ATP_Olambda__224,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: int] : aa_int_real(aTP_Lamm_sw(Uu),Uua) = suminf_real(aa_int_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_10426_ATP_Olambda__225,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: int] : aa_int_int(aTP_Lamm_ta(Uu),Uua) = suminf_int(aa_int_fun_nat_int(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_10427_ATP_Olambda__226,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_wz(Uu),Uua) = aa_complex_complex(cnj,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_10428_ATP_Olambda__227,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_hq(Uu),Uua)
    <=> ~ aa_Pro4592068669816069595_int_o(Uu,Uua) ) ).

% ATP.lambda_227
tff(fact_10429_ATP_Olambda__228,axiom,
    ! [Uu: fun_complex_o,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_hr(Uu),Uua)
    <=> ~ aa_complex_o(Uu,Uua) ) ).

% ATP.lambda_228
tff(fact_10430_ATP_Olambda__229,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_hs(Uu),Uua)
    <=> ~ aa_set_nat_o(Uu,Uua) ) ).

% ATP.lambda_229
tff(fact_10431_ATP_Olambda__230,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ht(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_230
tff(fact_10432_ATP_Olambda__231,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_hu(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_231
tff(fact_10433_ATP_Olambda__232,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vh(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_232
tff(fact_10434_ATP_Olambda__233,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_uo(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_233
tff(fact_10435_ATP_Olambda__234,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_hd(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_234
tff(fact_10436_ATP_Olambda__235,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_hb(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_235
tff(fact_10437_ATP_Olambda__236,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_hc(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_236
tff(fact_10438_ATP_Olambda__237,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rf(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_237
tff(fact_10439_ATP_Olambda__238,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rg(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_238
tff(fact_10440_ATP_Olambda__239,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rh(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_239
tff(fact_10441_ATP_Olambda__240,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ri(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_240
tff(fact_10442_ATP_Olambda__241,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_il(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_241
tff(fact_10443_ATP_Olambda__242,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ip(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_242
tff(fact_10444_ATP_Olambda__243,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_ig(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_243
tff(fact_10445_ATP_Olambda__244,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ij(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_244
tff(fact_10446_ATP_Olambda__245,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_im(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_245
tff(fact_10447_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ih(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_246
tff(fact_10448_ATP_Olambda__247,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_in(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_247
tff(fact_10449_ATP_Olambda__248,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ii(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_248
tff(fact_10450_ATP_Olambda__249,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ik(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_249
tff(fact_10451_ATP_Olambda__250,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_io(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_250
tff(fact_10452_ATP_Olambda__251,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_iv(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_251
tff(fact_10453_ATP_Olambda__252,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_iz(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_252
tff(fact_10454_ATP_Olambda__253,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_iq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_253
tff(fact_10455_ATP_Olambda__254,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_it(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_254
tff(fact_10456_ATP_Olambda__255,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_iw(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_255
tff(fact_10457_ATP_Olambda__256,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ir(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_256
tff(fact_10458_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qv(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_257
tff(fact_10459_ATP_Olambda__258,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ix(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_258
tff(fact_10460_ATP_Olambda__259,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qw(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_259
tff(fact_10461_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_qx(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_260
tff(fact_10462_ATP_Olambda__261,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_is(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_261
tff(fact_10463_ATP_Olambda__262,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_iu(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_262
tff(fact_10464_ATP_Olambda__263,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_iy(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_263
tff(fact_10465_ATP_Olambda__264,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_xh(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_264
tff(fact_10466_ATP_Olambda__265,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_xo(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_265
tff(fact_10467_ATP_Olambda__266,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_xg(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_266
tff(fact_10468_ATP_Olambda__267,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rb(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_267
tff(fact_10469_ATP_Olambda__268,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rc(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_268
tff(fact_10470_ATP_Olambda__269,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rd(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_269
tff(fact_10471_ATP_Olambda__270,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_re(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_270
tff(fact_10472_ATP_Olambda__271,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_mo(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_271
tff(fact_10473_ATP_Olambda__272,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_o,Uub: complex] :
      aa_complex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_ms(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_272
tff(fact_10474_ATP_Olambda__273,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_o,Uub: real] :
      aa_real_complex(aa_fun1098856492951973088omplex(aTP_Lamm_mj(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_273
tff(fact_10475_ATP_Olambda__274,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_mm(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_274
tff(fact_10476_ATP_Olambda__275,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_mp(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_275
tff(fact_10477_ATP_Olambda__276,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_o,Uub: nat] :
      aa_nat_complex(aa_fun1400689748890728928omplex(aTP_Lamm_mk(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_276
tff(fact_10478_ATP_Olambda__277,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_mq(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_277
tff(fact_10479_ATP_Olambda__278,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_o,Uub: int] :
      aa_int_complex(aa_fun6439487526662521568omplex(aTP_Lamm_ml(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_278
tff(fact_10480_ATP_Olambda__279,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_mn(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_279
tff(fact_10481_ATP_Olambda__280,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_mr(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_280
tff(fact_10482_ATP_Olambda__281,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_zv(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_281
tff(fact_10483_ATP_Olambda__282,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_jj(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_282
tff(fact_10484_ATP_Olambda__283,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: nat,Uub: complex] : aa_complex_complex(aa_nat3829984526574172007omplex(aTP_Lamm_sg(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(Uu,Uub),Uua) ).

% ATP.lambda_283
tff(fact_10485_ATP_Olambda__284,axiom,
    ! [Uu: fun_co8378349014452122532t_real,Uua: nat,Uub: complex] : aa_complex_real(aa_nat480766339200631653x_real(aTP_Lamm_rt(Uu),Uua),Uub) = aa_nat_real(aa_com20334259897932645t_real(Uu,Uub),Uua) ).

% ATP.lambda_284
tff(fact_10486_ATP_Olambda__285,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_mc(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_285
tff(fact_10487_ATP_Olambda__286,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ld(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_286
tff(fact_10488_ATP_Olambda__287,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_li(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_287
tff(fact_10489_ATP_Olambda__288,axiom,
    ! [Uu: fun_re2084383175013312036omplex,Uua: nat,Uub: real] : aa_real_complex(aa_nat7791439356240649701omplex(aTP_Lamm_ry(Uu),Uua),Uub) = aa_nat_complex(aa_rea1531280491672885477omplex(Uu,Uub),Uua) ).

% ATP.lambda_288
tff(fact_10490_ATP_Olambda__289,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_mf(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_289
tff(fact_10491_ATP_Olambda__290,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_lu(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_290
tff(fact_10492_ATP_Olambda__291,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_kv(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_291
tff(fact_10493_ATP_Olambda__292,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_lm(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_292
tff(fact_10494_ATP_Olambda__293,axiom,
    ! [Uu: fun_na1142814060743658952omplex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_sa(Uu),Uua),Uub) = aa_nat_complex(aa_nat6454746962644518025omplex(Uu,Uub),Uua) ).

% ATP.lambda_293
tff(fact_10495_ATP_Olambda__294,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jp(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_294
tff(fact_10496_ATP_Olambda__295,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_295
tff(fact_10497_ATP_Olambda__296,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_kz(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_296
tff(fact_10498_ATP_Olambda__297,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_lq(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_297
tff(fact_10499_ATP_Olambda__298,axiom,
    ! [Uu: fun_in8056368165505483812omplex,Uua: nat,Uub: int] : aa_int_complex(aa_nat7430282427850695397omplex(aTP_Lamm_sc(Uu),Uua),Uub) = aa_nat_complex(aa_int4144929030551567077omplex(Uu,Uub),Uua) ).

% ATP.lambda_298
tff(fact_10500_ATP_Olambda__299,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: nat,Uub: int] : aa_int_real(aa_nat_fun_int_real(aTP_Lamm_rw(Uu),Uua),Uub) = aa_nat_real(aa_int_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_299
tff(fact_10501_ATP_Olambda__300,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_ly(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_300
tff(fact_10502_ATP_Olambda__301,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_se(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_301
tff(fact_10503_ATP_Olambda__302,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_302
tff(fact_10504_ATP_Olambda__303,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_xc(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_303
tff(fact_10505_ATP_Olambda__304,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_xz((Uu)),Uua),(Uub)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_xy((Uu)),(Uub)),code_bit_cut_integer(Uua)) ).

% ATP.lambda_304
tff(fact_10506_ATP_Olambda__305,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wt(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ws(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_305
tff(fact_10507_ATP_Olambda__306,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wr(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wq(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_306
tff(fact_10508_ATP_Olambda__307,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wp(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wo(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_307
tff(fact_10509_ATP_Olambda__308,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wn(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wm(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_308
tff(fact_10510_ATP_Olambda__309,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_wl(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_wk(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_309
tff(fact_10511_ATP_Olambda__310,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_wj(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_wi(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_310
tff(fact_10512_ATP_Olambda__311,axiom,
    ! [Uu: set_complex,Uua: fun_co3258106519541737638omplex,Uub: nat] : aa_nat_complex(aa_fun5003373910569599327omplex(aTP_Lamm_sh(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(aTP_Lamm_sg(Uua),Uub),Uu) ).

% ATP.lambda_311
tff(fact_10513_ATP_Olambda__312,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_int,Uub: nat] : aa_nat_int(aa_fun8710705519803144921at_int(aTP_Lamm_sf(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_nat_fun_int_int(aTP_Lamm_se(Uua),Uub),Uu) ).

% ATP.lambda_312
tff(fact_10514_ATP_Olambda__313,axiom,
    ! [Uu: set_int,Uua: fun_in8056368165505483812omplex,Uub: nat] : aa_nat_complex(aa_fun3543561798013398749omplex(aTP_Lamm_sd(Uu),Uua),Uub) = groups3049146728041665814omplex(aa_nat7430282427850695397omplex(aTP_Lamm_sc(Uua),Uub),Uu) ).

% ATP.lambda_313
tff(fact_10515_ATP_Olambda__314,axiom,
    ! [Uu: set_nat,Uua: fun_na1142814060743658952omplex,Uub: nat] : aa_nat_complex(aa_fun4886358607025424513omplex(aTP_Lamm_sb(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_sa(Uua),Uub)),Uu) ).

% ATP.lambda_314
tff(fact_10516_ATP_Olambda__315,axiom,
    ! [Uu: set_real,Uua: fun_re2084383175013312036omplex,Uub: nat] : aa_nat_complex(aa_fun7632768522510439005omplex(aTP_Lamm_rz(Uu),Uua),Uub) = groups5754745047067104278omplex(aa_nat7791439356240649701omplex(aTP_Lamm_ry(Uua),Uub),Uu) ).

% ATP.lambda_315
tff(fact_10517_ATP_Olambda__316,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun2842512154077127769t_real(aTP_Lamm_rx(Uu),Uua),Uub) = groups8778361861064173332t_real(aa_nat_fun_int_real(aTP_Lamm_rw(Uua),Uub),Uu) ).

% ATP.lambda_316
tff(fact_10518_ATP_Olambda__317,axiom,
    ! [Uu: set_complex,Uua: fun_co8378349014452122532t_real,Uub: nat] : aa_nat_real(aa_fun471735231769099483t_real(aTP_Lamm_ru(Uu),Uua),Uub) = groups5808333547571424918x_real(aa_nat480766339200631653x_real(aTP_Lamm_rt(Uua),Uub),Uu) ).

% ATP.lambda_317
tff(fact_10519_ATP_Olambda__318,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: nat] : aa_nat_real(aa_fun628238740774635481t_real(aTP_Lamm_rv(Uu),Uua),Uub) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_mf(Uua),Uub),Uu) ).

% ATP.lambda_318
tff(fact_10520_ATP_Olambda__319,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_jq(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jp(Uu),Uub)),Uua) ).

% ATP.lambda_319
tff(fact_10521_ATP_Olambda__320,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_sj(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jp(Uua),Uub)),Uu) ).

% ATP.lambda_320
tff(fact_10522_ATP_Olambda__321,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_jn(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jm(Uu),Uub)),Uua) ).

% ATP.lambda_321
tff(fact_10523_ATP_Olambda__322,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_si(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jm(Uua),Uub)),Uu) ).

% ATP.lambda_322
tff(fact_10524_ATP_Olambda__323,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_jk(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_jj(Uu),Uub),Uua) ).

% ATP.lambda_323
tff(fact_10525_ATP_Olambda__324,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_jh(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_jg(Uu),Uub),Uua) ).

% ATP.lambda_324
tff(fact_10526_ATP_Olambda__325,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ym(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_325
tff(fact_10527_ATP_Olambda__326,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_uw(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_326
tff(fact_10528_ATP_Olambda__327,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_ux(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_327
tff(fact_10529_ATP_Olambda__328,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_uy(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_328
tff(fact_10530_ATP_Olambda__329,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_uz(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_329
tff(fact_10531_ATP_Olambda__330,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_va(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_330
tff(fact_10532_ATP_Olambda__331,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oa(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_331
tff(fact_10533_ATP_Olambda__332,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oc(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_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_332
tff(fact_10534_ATP_Olambda__333,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ob(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_333
tff(fact_10535_ATP_Olambda__334,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yj(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_334
tff(fact_10536_ATP_Olambda__335,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vb(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_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_335
tff(fact_10537_ATP_Olambda__336,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_df(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_336
tff(fact_10538_ATP_Olambda__337,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_de(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_337
tff(fact_10539_ATP_Olambda__338,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_dg(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_338
tff(fact_10540_ATP_Olambda__339,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_dd(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_339
tff(fact_10541_ATP_Olambda__340,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_dh(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_340
tff(fact_10542_ATP_Olambda__341,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_da(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_341
tff(fact_10543_ATP_Olambda__342,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_cz(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_342
tff(fact_10544_ATP_Olambda__343,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_db(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_343
tff(fact_10545_ATP_Olambda__344,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_cy(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_344
tff(fact_10546_ATP_Olambda__345,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_dc(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_345
tff(fact_10547_ATP_Olambda__346,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_xe(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_346
tff(fact_10548_ATP_Olambda__347,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_xi(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_347
tff(fact_10549_ATP_Olambda__348,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_xf(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_348
tff(fact_10550_ATP_Olambda__349,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vm(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_349
tff(fact_10551_ATP_Olambda__350,axiom,
    ! [Uu: vEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_bw(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_350
tff(fact_10552_ATP_Olambda__351,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_by(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_351
tff(fact_10553_ATP_Olambda__352,axiom,
    ! [Uu: complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_bz(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_352
tff(fact_10554_ATP_Olambda__353,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_353
tff(fact_10555_ATP_Olambda__354,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_ca(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_354
tff(fact_10556_ATP_Olambda__355,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_cb(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_355
tff(fact_10557_ATP_Olambda__356,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_cc(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_356
tff(fact_10558_ATP_Olambda__357,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hw(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_357
tff(fact_10559_ATP_Olambda__358,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dz(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_358
tff(fact_10560_ATP_Olambda__359,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hx(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_359
tff(fact_10561_ATP_Olambda__360,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_eb(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_360
tff(fact_10562_ATP_Olambda__361,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ec(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_361
tff(fact_10563_ATP_Olambda__362,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ea(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_362
tff(fact_10564_ATP_Olambda__363,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ak(Uu),Uua),Uub)
    <=> ( aa_nat_o(vEBT_vebt_member(Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_363
tff(fact_10565_ATP_Olambda__364,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_al(Uu),Uua),Uub)
    <=> ( aa_nat_o(vEBT_vebt_member(Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_364
tff(fact_10566_ATP_Olambda__365,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_xn(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_365
tff(fact_10567_ATP_Olambda__366,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_xm(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_366
tff(fact_10568_ATP_Olambda__367,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_bd(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_367
tff(fact_10569_ATP_Olambda__368,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_be(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_368
tff(fact_10570_ATP_Olambda__369,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bc(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_369
tff(fact_10571_ATP_Olambda__370,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bf(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_370
tff(fact_10572_ATP_Olambda__371,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bg(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_371
tff(fact_10573_ATP_Olambda__372,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_372
tff(fact_10574_ATP_Olambda__373,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_cd(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_373
tff(fact_10575_ATP_Olambda__374,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_ce(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_374
tff(fact_10576_ATP_Olambda__375,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_cf(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_375
tff(fact_10577_ATP_Olambda__376,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_cj(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_376
tff(fact_10578_ATP_Olambda__377,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_377
tff(fact_10579_ATP_Olambda__378,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ch(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_378
tff(fact_10580_ATP_Olambda__379,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ci(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_379
tff(fact_10581_ATP_Olambda__380,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_380
tff(fact_10582_ATP_Olambda__381,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_cl(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_381
tff(fact_10583_ATP_Olambda__382,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_cm(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_382
tff(fact_10584_ATP_Olambda__383,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_cq(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_383
tff(fact_10585_ATP_Olambda__384,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_cn(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_384
tff(fact_10586_ATP_Olambda__385,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_385
tff(fact_10587_ATP_Olambda__386,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cp(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_386
tff(fact_10588_ATP_Olambda__387,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_fo(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_387
tff(fact_10589_ATP_Olambda__388,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_fw(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_388
tff(fact_10590_ATP_Olambda__389,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_fi(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_389
tff(fact_10591_ATP_Olambda__390,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_fq(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_390
tff(fact_10592_ATP_Olambda__391,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_fy(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_391
tff(fact_10593_ATP_Olambda__392,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_fk(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_392
tff(fact_10594_ATP_Olambda__393,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_fs(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_393
tff(fact_10595_ATP_Olambda__394,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_ga(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_394
tff(fact_10596_ATP_Olambda__395,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_fm(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_395
tff(fact_10597_ATP_Olambda__396,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_fu(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_396
tff(fact_10598_ATP_Olambda__397,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_eu(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_397
tff(fact_10599_ATP_Olambda__398,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_fc(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_398
tff(fact_10600_ATP_Olambda__399,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_eo(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_399
tff(fact_10601_ATP_Olambda__400,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_ew(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_400
tff(fact_10602_ATP_Olambda__401,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_fe(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_401
tff(fact_10603_ATP_Olambda__402,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_eq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_402
tff(fact_10604_ATP_Olambda__403,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_ey(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_403
tff(fact_10605_ATP_Olambda__404,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_fg(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_404
tff(fact_10606_ATP_Olambda__405,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_es(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_405
tff(fact_10607_ATP_Olambda__406,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_fa(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_406
tff(fact_10608_ATP_Olambda__407,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & ~ member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_407
tff(fact_10609_ATP_Olambda__408,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ~ aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_408
tff(fact_10610_ATP_Olambda__409,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ~ aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_409
tff(fact_10611_ATP_Olambda__410,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & ~ aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_410
tff(fact_10612_ATP_Olambda__411,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_av(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_411
tff(fact_10613_ATP_Olambda__412,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_au(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_412
tff(fact_10614_ATP_Olambda__413,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wf(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_413
tff(fact_10615_ATP_Olambda__414,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_aaq(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu) ) ).

% ATP.lambda_414
tff(fact_10616_ATP_Olambda__415,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aas(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V975177566351809787t_real(Uua),Uub)),Uu) ) ).

% ATP.lambda_415
tff(fact_10617_ATP_Olambda__416,axiom,
    ! [Uu: set_Pr448751882837621926eger_o,Uua: code_integer,Uub: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aTP_Lamm_hy(Uu),Uua),(Uub))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(Uua,(Uub)),Uu) ) ).

% ATP.lambda_416
tff(fact_10618_ATP_Olambda__417,axiom,
    ! [Uu: set_Pr8218934625190621173um_num,Uua: num,Uub: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_hz(Uu),Uua),Uub)
    <=> member7279096912039735102um_num(product_Pair_num_num(Uua,Uub),Uu) ) ).

% ATP.lambda_417
tff(fact_10619_ATP_Olambda__418,axiom,
    ! [Uu: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
      ( aa_num_o(aa_nat_fun_num_o(aTP_Lamm_ia(Uu),Uua),Uub)
    <=> member9148766508732265716at_num(product_Pair_nat_num(Uua,Uub),Uu) ) ).

% ATP.lambda_418
tff(fact_10620_ATP_Olambda__419,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ib(Uu),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_419
tff(fact_10621_ATP_Olambda__420,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ic(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu) ) ).

% ATP.lambda_420
tff(fact_10622_ATP_Olambda__421,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_en(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_421
tff(fact_10623_ATP_Olambda__422,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_wb(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_422
tff(fact_10624_ATP_Olambda__423,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tf(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_423
tff(fact_10625_ATP_Olambda__424,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tg(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_424
tff(fact_10626_ATP_Olambda__425,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_zl(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_425
tff(fact_10627_ATP_Olambda__426,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yk(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_426
tff(fact_10628_ATP_Olambda__427,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ss(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_427
tff(fact_10629_ATP_Olambda__428,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sq(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_428
tff(fact_10630_ATP_Olambda__429,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vp(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_429
tff(fact_10631_ATP_Olambda__430,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cw(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_430
tff(fact_10632_ATP_Olambda__431,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aad(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_431
tff(fact_10633_ATP_Olambda__432,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zw(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_432
tff(fact_10634_ATP_Olambda__433,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_wv(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_433
tff(fact_10635_ATP_Olambda__434,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_nn(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_434
tff(fact_10636_ATP_Olambda__435,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_nr(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).

% ATP.lambda_435
tff(fact_10637_ATP_Olambda__436,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_nu(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).

% ATP.lambda_436
tff(fact_10638_ATP_Olambda__437,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_no(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_437
tff(fact_10639_ATP_Olambda__438,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ns(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_438
tff(fact_10640_ATP_Olambda__439,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nv(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_439
tff(fact_10641_ATP_Olambda__440,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_np(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_440
tff(fact_10642_ATP_Olambda__441,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_nw(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_441
tff(fact_10643_ATP_Olambda__442,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_nq(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_442
tff(fact_10644_ATP_Olambda__443,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_nt(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_443
tff(fact_10645_ATP_Olambda__444,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_km(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_444
tff(fact_10646_ATP_Olambda__445,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rr(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_445
tff(fact_10647_ATP_Olambda__446,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kn(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_446
tff(fact_10648_ATP_Olambda__447,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kl(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_447
tff(fact_10649_ATP_Olambda__448,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_od(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu,Uub)) ).

% ATP.lambda_448
tff(fact_10650_ATP_Olambda__449,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_oe(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_449
tff(fact_10651_ATP_Olambda__450,axiom,
    ! [Uu: fun_o_nat,Uua: fun_o_nat,Uub: $o] : aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_tl(Uu),Uua),(Uub)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_o_nat(Uua,(Uub))),aa_o_nat(Uu,(Uub))) ).

% ATP.lambda_450
tff(fact_10652_ATP_Olambda__451,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_of(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu,Uub)) ).

% ATP.lambda_451
tff(fact_10653_ATP_Olambda__452,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oh(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_452
tff(fact_10654_ATP_Olambda__453,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_og(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_453
tff(fact_10655_ATP_Olambda__454,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ki(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_454
tff(fact_10656_ATP_Olambda__455,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ro(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_455
tff(fact_10657_ATP_Olambda__456,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kk(Uu),Uua),Uub) = aa_real_real(plus_plus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_456
tff(fact_10658_ATP_Olambda__457,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kj(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_457
tff(fact_10659_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_rp(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_458
tff(fact_10660_ATP_Olambda__459,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kh(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_459
tff(fact_10661_ATP_Olambda__460,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yi(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_460
tff(fact_10662_ATP_Olambda__461,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_du(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        | aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_461
tff(fact_10663_ATP_Olambda__462,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_dy(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_462
tff(fact_10664_ATP_Olambda__463,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_dv(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_463
tff(fact_10665_ATP_Olambda__464,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_dw(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_464
tff(fact_10666_ATP_Olambda__465,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_dx(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_465
tff(fact_10667_ATP_Olambda__466,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_dp(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_466
tff(fact_10668_ATP_Olambda__467,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_dt(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_467
tff(fact_10669_ATP_Olambda__468,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_dq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_468
tff(fact_10670_ATP_Olambda__469,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_dr(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_469
tff(fact_10671_ATP_Olambda__470,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ds(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_470
tff(fact_10672_ATP_Olambda__471,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_ji(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_471
tff(fact_10673_ATP_Olambda__472,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_jo(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_472
tff(fact_10674_ATP_Olambda__473,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_jl(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_473
tff(fact_10675_ATP_Olambda__474,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_jf(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_474
tff(fact_10676_ATP_Olambda__475,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cx(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_475
tff(fact_10677_ATP_Olambda__476,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mi(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_476
tff(fact_10678_ATP_Olambda__477,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_mh(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_477
tff(fact_10679_ATP_Olambda__478,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ko(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_478
tff(fact_10680_ATP_Olambda__479,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ra(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_479
tff(fact_10681_ATP_Olambda__480,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kp(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_480
tff(fact_10682_ATP_Olambda__481,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ka(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_481
tff(fact_10683_ATP_Olambda__482,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_rn(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_482
tff(fact_10684_ATP_Olambda__483,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kc(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_483
tff(fact_10685_ATP_Olambda__484,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_484
tff(fact_10686_ATP_Olambda__485,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jz(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_485
tff(fact_10687_ATP_Olambda__486,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_tr(Uu),Uua),Uub) = minus_8373710615458151222nteger(aa_nat_Code_integer(Uu,Uub),Uua) ).

% ATP.lambda_486
tff(fact_10688_ATP_Olambda__487,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ts(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_487
tff(fact_10689_ATP_Olambda__488,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_tp(Uu),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uub)),Uua) ).

% ATP.lambda_488
tff(fact_10690_ATP_Olambda__489,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_tq(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_489
tff(fact_10691_ATP_Olambda__490,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yf(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_490
tff(fact_10692_ATP_Olambda__491,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zq(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_491
tff(fact_10693_ATP_Olambda__492,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_yh(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_492
tff(fact_10694_ATP_Olambda__493,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_493
tff(fact_10695_ATP_Olambda__494,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_br(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_494
tff(fact_10696_ATP_Olambda__495,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_bs(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_495
tff(fact_10697_ATP_Olambda__496,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_496
tff(fact_10698_ATP_Olambda__497,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_497
tff(fact_10699_ATP_Olambda__498,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_498
tff(fact_10700_ATP_Olambda__499,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bv(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_499
tff(fact_10701_ATP_Olambda__500,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_tt(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_500
tff(fact_10702_ATP_Olambda__501,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_px(Uu),Uua),Uub) = plus_p5714425477246183910nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_501
tff(fact_10703_ATP_Olambda__502,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pu(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_502
tff(fact_10704_ATP_Olambda__503,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pz(Uu),Uua),Uub) = aa_real_real(plus_plus_real(Uu),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_503
tff(fact_10705_ATP_Olambda__504,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_pv(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_504
tff(fact_10706_ATP_Olambda__505,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_py(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_505
tff(fact_10707_ATP_Olambda__506,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_pw(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_506
tff(fact_10708_ATP_Olambda__507,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_uk(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_507
tff(fact_10709_ATP_Olambda__508,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_un(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_508
tff(fact_10710_ATP_Olambda__509,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ul(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_509
tff(fact_10711_ATP_Olambda__510,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_um(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_510
tff(fact_10712_ATP_Olambda__511,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zf(Uu),Uua),Uub) = aa_real_real(divide_divide_real(Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_511
tff(fact_10713_ATP_Olambda__512,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_om(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_512
tff(fact_10714_ATP_Olambda__513,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_op(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_513
tff(fact_10715_ATP_Olambda__514,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_on(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_514
tff(fact_10716_ATP_Olambda__515,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oo(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_515
tff(fact_10717_ATP_Olambda__516,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pi(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_516
tff(fact_10718_ATP_Olambda__517,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aab(Uu),Uua),Uub)
    <=> aa_set_real_o(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_517
tff(fact_10719_ATP_Olambda__518,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ke(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_518
tff(fact_10720_ATP_Olambda__519,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qz(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_519
tff(fact_10721_ATP_Olambda__520,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kg(Uu),Uua),Uub) = aa_real_real(times_times_real(Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_520
tff(fact_10722_ATP_Olambda__521,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kf(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_521
tff(fact_10723_ATP_Olambda__522,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kd(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_522
tff(fact_10724_ATP_Olambda__523,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_rl(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uua),aa_nat_complex(Uu,Uub)) ).

% ATP.lambda_523
tff(fact_10725_ATP_Olambda__524,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rm(Uu),Uua),Uub) = aa_real_real(times_times_real(Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_524
tff(fact_10726_ATP_Olambda__525,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tk(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_525
tff(fact_10727_ATP_Olambda__526,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tj(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_526
tff(fact_10728_ATP_Olambda__527,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_zz(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(plus_plus_real(Uub),Uua)) ) ).

% ATP.lambda_527
tff(fact_10729_ATP_Olambda__528,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qy(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_528
tff(fact_10730_ATP_Olambda__529,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ol(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_529
tff(fact_10731_ATP_Olambda__530,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zy(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_530
tff(fact_10732_ATP_Olambda__531,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ok(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_531
tff(fact_10733_ATP_Olambda__532,axiom,
    ! [Uu: fun_Pr2118491707810114563at_nat,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gw(Uu),Uua),Uub) = aa_Pro262675715343573244at_nat(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_532
tff(fact_10734_ATP_Olambda__533,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gx(Uu),Uua),Uub)
    <=> aa_Pro6150637604445964947_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ) ).

% ATP.lambda_533
tff(fact_10735_ATP_Olambda__534,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gy(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_534
tff(fact_10736_ATP_Olambda__535,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_gz(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_535
tff(fact_10737_ATP_Olambda__536,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ha(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_536
tff(fact_10738_ATP_Olambda__537,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_st(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub))) ).

% ATP.lambda_537
tff(fact_10739_ATP_Olambda__538,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sr(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_538
tff(fact_10740_ATP_Olambda__539,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zk(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_539
tff(fact_10741_ATP_Olambda__540,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_nd(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_real(Uua,Uuc),aa_VEBT_VEBT_real(Uub,Uuc)) ).

% ATP.lambda_540
tff(fact_10742_ATP_Olambda__541,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_nf(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_rat(Uua,Uuc),aa_VEBT_VEBT_rat(Uub,Uuc)) ).

% ATP.lambda_541
tff(fact_10743_ATP_Olambda__542,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_nh(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_nat(Uua,Uuc),aa_VEBT_VEBT_nat(Uub,Uuc)) ).

% ATP.lambda_542
tff(fact_10744_ATP_Olambda__543,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_nj(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_int(Uua,Uuc),aa_VEBT_VEBT_int(Uub,Uuc)) ).

% ATP.lambda_543
tff(fact_10745_ATP_Olambda__544,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_ne(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_544
tff(fact_10746_ATP_Olambda__545,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_ng(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_545
tff(fact_10747_ATP_Olambda__546,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ni(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_546
tff(fact_10748_ATP_Olambda__547,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] :
      aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_nk(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_int(Uua,Uuc),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_547
tff(fact_10749_ATP_Olambda__548,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_nl(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_real(Uua,Uuc),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_548
tff(fact_10750_ATP_Olambda__549,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_nm(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_549
tff(fact_10751_ATP_Olambda__550,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qu(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qt(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_550
tff(fact_10752_ATP_Olambda__551,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qs(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qr(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_551
tff(fact_10753_ATP_Olambda__552,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qq(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qp(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_552
tff(fact_10754_ATP_Olambda__553,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_xy((Uu)),(Uua)),Uub),(Uuc)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_xx((Uu)),(Uua)),(Uuc)),code_bit_cut_integer(Uub)) ).

% ATP.lambda_553
tff(fact_10755_ATP_Olambda__554,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_jy(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_jx(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_554
tff(fact_10756_ATP_Olambda__555,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_jw(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_jv(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_555
tff(fact_10757_ATP_Olambda__556,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_ju(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_jt(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_556
tff(fact_10758_ATP_Olambda__557,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_js(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_jr(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_557
tff(fact_10759_ATP_Olambda__558,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_mg(Uu),Uua),Uub),Uuc) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_mf(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_ed(Uu),Uub),Uuc))) ).

% ATP.lambda_558
tff(fact_10760_ATP_Olambda__559,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_md(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_mc(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_em(Uu),Uub),Uuc))) ).

% ATP.lambda_559
tff(fact_10761_ATP_Olambda__560,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_lz(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_ly(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_ej(Uu),Uub),Uuc))) ).

% ATP.lambda_560
tff(fact_10762_ATP_Olambda__561,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_lv(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_lu(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_ed(Uu),Uub),Uuc))) ).

% ATP.lambda_561
tff(fact_10763_ATP_Olambda__562,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: complex] : aa_complex_complex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_lr(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_lq(Uua),Uuc),collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_el(Uu),Uub),Uuc))) ).

% ATP.lambda_562
tff(fact_10764_ATP_Olambda__563,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : aa_complex_complex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_ln(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_lm(Uua),Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_ei(Uu),Uub),Uuc))) ).

% ATP.lambda_563
tff(fact_10765_ATP_Olambda__564,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: complex] : aa_complex_complex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_lj(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_li(Uua),Uuc),collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_ef(Uu),Uub),Uuc))) ).

% ATP.lambda_564
tff(fact_10766_ATP_Olambda__565,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_lf(Uu),Uua),Uub),Uuc) = groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ld(Uua),Uuc),collect_complex(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_le(Uu),Uub),Uuc))) ).

% ATP.lambda_565
tff(fact_10767_ATP_Olambda__566,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_la(Uu),Uua),Uub),Uuc) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_kz(Uua),Uuc),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_eh(Uu),Uub),Uuc))) ).

% ATP.lambda_566
tff(fact_10768_ATP_Olambda__567,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_kw(Uu),Uua),Uub),Uuc) = groups1932886352136224148al_int(aa_int_fun_real_int(aTP_Lamm_kv(Uua),Uuc),collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_ee(Uu),Uub),Uuc))) ).

% ATP.lambda_567
tff(fact_10769_ATP_Olambda__568,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_vr(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_vq(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_568
tff(fact_10770_ATP_Olambda__569,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yr(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_569
tff(fact_10771_ATP_Olambda__570,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yp(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_570
tff(fact_10772_ATP_Olambda__571,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yo(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_571
tff(fact_10773_ATP_Olambda__572,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yn(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_572
tff(fact_10774_ATP_Olambda__573,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tu(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uub)) ).

% ATP.lambda_573
tff(fact_10775_ATP_Olambda__574,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ua(Uu),Uua),Uub),Uuc) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_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_574
tff(fact_10776_ATP_Olambda__575,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tw(Uu),Uua),Uub),Uuc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uu),Uub)) ).

% ATP.lambda_575
tff(fact_10777_ATP_Olambda__576,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ty(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_576
tff(fact_10778_ATP_Olambda__577,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qp(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uua),Uu)),Uuc)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_577
tff(fact_10779_ATP_Olambda__578,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qt(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_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_578
tff(fact_10780_ATP_Olambda__579,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qr(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_579
tff(fact_10781_ATP_Olambda__580,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vo(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_580
tff(fact_10782_ATP_Olambda__581,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_uc(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc)) ).

% ATP.lambda_581
tff(fact_10783_ATP_Olambda__582,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_uf(Uu),Uua),Uub),Uuc) = aa_real_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_582
tff(fact_10784_ATP_Olambda__583,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ud(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_583
tff(fact_10785_ATP_Olambda__584,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ue(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_584
tff(fact_10786_ATP_Olambda__585,axiom,
    ! [Uu: set_complex,Uua: fun_re7319081385389557872plex_o,Uub: real,Uuc: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_lg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_585
tff(fact_10787_ATP_Olambda__586,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_lk(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_586
tff(fact_10788_ATP_Olambda__587,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_lo(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_587
tff(fact_10789_ATP_Olambda__588,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_ma(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_588
tff(fact_10790_ATP_Olambda__589,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_ls(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_589
tff(fact_10791_ATP_Olambda__590,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_lw(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_590
tff(fact_10792_ATP_Olambda__591,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_lb(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_591
tff(fact_10793_ATP_Olambda__592,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_kt(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_592
tff(fact_10794_ATP_Olambda__593,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_kx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_593
tff(fact_10795_ATP_Olambda__594,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_em(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_594
tff(fact_10796_ATP_Olambda__595,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_le(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_595
tff(fact_10797_ATP_Olambda__596,axiom,
    ! [Uu: set_real,Uua: fun_re7319081385389557872plex_o,Uub: complex,Uuc: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_ef(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_596
tff(fact_10798_ATP_Olambda__597,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_ed(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_597
tff(fact_10799_ATP_Olambda__598,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_ee(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_598
tff(fact_10800_ATP_Olambda__599,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_ei(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_599
tff(fact_10801_ATP_Olambda__600,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_eg(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_600
tff(fact_10802_ATP_Olambda__601,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_eh(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_601
tff(fact_10803_ATP_Olambda__602,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_el(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_602
tff(fact_10804_ATP_Olambda__603,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_ej(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_603
tff(fact_10805_ATP_Olambda__604,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ek(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_604
tff(fact_10806_ATP_Olambda__605,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tv(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_605
tff(fact_10807_ATP_Olambda__606,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ub(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_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_606
tff(fact_10808_ATP_Olambda__607,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tx(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_607
tff(fact_10809_ATP_Olambda__608,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_tz(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_608
tff(fact_10810_ATP_Olambda__609,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ug(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_609
tff(fact_10811_ATP_Olambda__610,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_uj(Uu),Uua),Uub),Uuc) = aa_real_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_610
tff(fact_10812_ATP_Olambda__611,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_uh(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_611
tff(fact_10813_ATP_Olambda__612,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ui(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_612
tff(fact_10814_ATP_Olambda__613,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vn(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_613
tff(fact_10815_ATP_Olambda__614,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_wk(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_614
tff(fact_10816_ATP_Olambda__615,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_wi(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_615
tff(fact_10817_ATP_Olambda__616,axiom,
    ! [Uu: set_complex,Uua: complex,Uub: complex,Uuc: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aa_com7221863818706917250plex_o(aTP_Lamm_aau(Uu),Uua),Uub),Uuc)
    <=> ( ( Uub = Uua )
       => aa_set_complex_o(member_complex(Uuc),Uu) ) ) ).

% ATP.lambda_616
tff(fact_10818_ATP_Olambda__617,axiom,
    ! [Uu: set_real,Uua: real,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_rea1652550732481586312real_o(aTP_Lamm_aav(Uu),Uua),Uub),Uuc)
    <=> ( ( Uub = Uua )
       => aa_set_real_o(member_real(Uuc),Uu) ) ) ).

% ATP.lambda_617
tff(fact_10819_ATP_Olambda__618,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_ev(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_618
tff(fact_10820_ATP_Olambda__619,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_fd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(times_times_real(aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_619
tff(fact_10821_ATP_Olambda__620,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_ep(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_620
tff(fact_10822_ATP_Olambda__621,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_ex(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(times_times_real(aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_621
tff(fact_10823_ATP_Olambda__622,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_ff(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_622
tff(fact_10824_ATP_Olambda__623,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_er(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_623
tff(fact_10825_ATP_Olambda__624,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_ez(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(times_times_real(aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_624
tff(fact_10826_ATP_Olambda__625,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_fh(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_625
tff(fact_10827_ATP_Olambda__626,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_et(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_626
tff(fact_10828_ATP_Olambda__627,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_fb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(times_times_real(aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_627
tff(fact_10829_ATP_Olambda__628,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_fp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_628
tff(fact_10830_ATP_Olambda__629,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_fx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(plus_plus_real(aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_629
tff(fact_10831_ATP_Olambda__630,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_fj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_630
tff(fact_10832_ATP_Olambda__631,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_fr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(plus_plus_real(aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_631
tff(fact_10833_ATP_Olambda__632,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_fz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_632
tff(fact_10834_ATP_Olambda__633,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_fl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_633
tff(fact_10835_ATP_Olambda__634,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_ft(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(plus_plus_real(aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_634
tff(fact_10836_ATP_Olambda__635,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_gb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_635
tff(fact_10837_ATP_Olambda__636,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_fn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_636
tff(fact_10838_ATP_Olambda__637,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_fv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(plus_plus_real(aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_637
tff(fact_10839_ATP_Olambda__638,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_th(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uuc)) ).

% ATP.lambda_638
tff(fact_10840_ATP_Olambda__639,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ti(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_639
tff(fact_10841_ATP_Olambda__640,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_vq(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_640
tff(fact_10842_ATP_Olambda__641,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_jt(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_641
tff(fact_10843_ATP_Olambda__642,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_jx(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_642
tff(fact_10844_ATP_Olambda__643,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_jv(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_643
tff(fact_10845_ATP_Olambda__644,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_jr(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_644
tff(fact_10846_ATP_Olambda__645,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_mb(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_ma(Uu),Uub),Uuc))) ).

% ATP.lambda_645
tff(fact_10847_ATP_Olambda__646,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_lx(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_lw(Uu),Uub),Uuc))) ).

% ATP.lambda_646
tff(fact_10848_ATP_Olambda__647,axiom,
    ! [Uu: set_nat,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_me(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_ls(Uu),Uub),Uuc))) ).

% ATP.lambda_647
tff(fact_10849_ATP_Olambda__648,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_lt(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_ls(Uu),Uub),Uuc))) ).

% ATP.lambda_648
tff(fact_10850_ATP_Olambda__649,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: int] : aa_int_complex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_lp(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_lo(Uu),Uub),Uuc))) ).

% ATP.lambda_649
tff(fact_10851_ATP_Olambda__650,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : aa_nat_complex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_ll(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_lk(Uu),Uub),Uuc))) ).

% ATP.lambda_650
tff(fact_10852_ATP_Olambda__651,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: real] : aa_real_complex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_lh(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_lg(Uu),Uub),Uuc))) ).

% ATP.lambda_651
tff(fact_10853_ATP_Olambda__652,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_lc(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_lb(Uu),Uub),Uuc))) ).

% ATP.lambda_652
tff(fact_10854_ATP_Olambda__653,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_ky(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_kx(Uu),Uub),Uuc))) ).

% ATP.lambda_653
tff(fact_10855_ATP_Olambda__654,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_ku(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_kt(Uu),Uub),Uuc))) ).

% ATP.lambda_654
tff(fact_10856_ATP_Olambda__655,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_hg((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_655
tff(fact_10857_ATP_Olambda__656,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_hh((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_656
tff(fact_10858_ATP_Olambda__657,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_gr(Uu),Uua),Uub),Uuc)
    <=> aa_Pro6150637604445964947_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Uua,Uub),Uuc)) ) ).

% ATP.lambda_657
tff(fact_10859_ATP_Olambda__658,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_gt(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_658
tff(fact_10860_ATP_Olambda__659,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_gu(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_659
tff(fact_10861_ATP_Olambda__660,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_gs(Uu),Uua),Uub),Uuc) = aa_o_P6312935399938478757at_nat(Uu,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_660
tff(fact_10862_ATP_Olambda__661,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_gv(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_661
tff(fact_10863_ATP_Olambda__662,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_gm(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_662
tff(fact_10864_ATP_Olambda__663,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_gn(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_663
tff(fact_10865_ATP_Olambda__664,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_go(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_664
tff(fact_10866_ATP_Olambda__665,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_gp(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_665
tff(fact_10867_ATP_Olambda__666,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_gq(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_666
tff(fact_10868_ATP_Olambda__667,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_or(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_667
tff(fact_10869_ATP_Olambda__668,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oq(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_668
tff(fact_10870_ATP_Olambda__669,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wq(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_669
tff(fact_10871_ATP_Olambda__670,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wm(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_670
tff(fact_10872_ATP_Olambda__671,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ws(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_671
tff(fact_10873_ATP_Olambda__672,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wo(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_672
tff(fact_10874_ATP_Olambda__673,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_xx((Uu)),(Uua)),(Uub)),Uuc),(Uud)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_xw((Uu)),(Uua)),(Uub)),(Uud)),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_673
tff(fact_10875_ATP_Olambda__674,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_yq(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yp(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(times_times_real(Uub),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_674
tff(fact_10876_ATP_Olambda__675,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_xw((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_xv((Uu)),(Uua)),(Uub)),(Uuc)),(Uue)),code_bit_cut_integer(Uud)) ).

% ATP.lambda_675
tff(fact_10877_ATP_Olambda__676,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_xv((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_xu((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf)),code_bit_cut_integer(Uue)) ).

% ATP.lambda_676
tff(fact_10878_ATP_Olambda__677,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_aao(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))
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I5)),X_13)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I5) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Uua))
             => ~ ? [X6: nat] : aa_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_677
tff(fact_10879_ATP_Olambda__678,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_xt((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_678
tff(fact_10880_ATP_Olambda__679,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_xu((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_xt((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug)),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_679
tff(fact_10881_ATP_Olambda__680,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_rj,Uu),Uua) = Uu ).

% ATP.lambda_680
tff(fact_10882_ATP_Olambda__681,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rk,Uu),Uua) = Uu ).

% ATP.lambda_681
tff(fact_10883_ATP_Olambda__682,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hi,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_682
tff(fact_10884_ATP_Olambda__683,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hj,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_683
tff(fact_10885_ATP_Olambda__684,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_af,Uu) = Uu ).

% ATP.lambda_684
tff(fact_10886_ATP_Olambda__685,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ag,Uu) = Uu ).

% ATP.lambda_685
tff(fact_10887_ATP_Olambda__686,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ah,Uu) = Uu ).

% ATP.lambda_686
tff(fact_10888_ATP_Olambda__687,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ai,Uu) = Uu ).

% ATP.lambda_687
tff(fact_10889_ATP_Olambda__688,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aj,Uu) = Uu ).

% ATP.lambda_688
tff(fact_10890_ATP_Olambda__689,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_cr,Uu) = zero_zero_complex ).

% ATP.lambda_689
tff(fact_10891_ATP_Olambda__690,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_cs,Uu) = zero_zero_real ).

% ATP.lambda_690
tff(fact_10892_ATP_Olambda__691,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ct,Uu) = zero_zero_rat ).

% ATP.lambda_691
tff(fact_10893_ATP_Olambda__692,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_qm,Uu) = zero_zero_complex ).

% ATP.lambda_692
tff(fact_10894_ATP_Olambda__693,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_id,Uu) = zero_zero_real ).

% ATP.lambda_693
tff(fact_10895_ATP_Olambda__694,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cu,Uu) = zero_zero_nat ).

% ATP.lambda_694
tff(fact_10896_ATP_Olambda__695,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_qn,Uu) = zero_zero_int ).

% ATP.lambda_695
tff(fact_10897_ATP_Olambda__696,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cv,Uu) = zero_zero_int ).

% ATP.lambda_696
tff(fact_10898_ATP_Olambda__697,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aw,Uu)
    <=> $false ) ).

% ATP.lambda_697
tff(fact_10899_ATP_Olambda__698,axiom,
    ! [Uu: complex] :
      ( aa_complex_o(aTP_Lamm_ax,Uu)
    <=> $false ) ).

% ATP.lambda_698
tff(fact_10900_ATP_Olambda__699,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_bb,Uu)
    <=> $false ) ).

% ATP.lambda_699
tff(fact_10901_ATP_Olambda__700,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ay,Uu)
    <=> $false ) ).

% ATP.lambda_700
tff(fact_10902_ATP_Olambda__701,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_az,Uu)
    <=> $false ) ).

% ATP.lambda_701
tff(fact_10903_ATP_Olambda__702,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_ba,Uu)
    <=> $false ) ).

% ATP.lambda_702
tff(fact_10904_ATP_Olambda__703,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_xj,Uu)
    <=> $true ) ).

% ATP.lambda_703

% Helper facts (15)
tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X7: int,Y3: int] :
      ( ( X7 != Y3 )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X7),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X7: int,Y3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X7),Y3)
      | ( X7 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X7: nat,Y3: nat] :
      ( ( X7 != Y3 )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X7),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X7: nat,Y3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X7),Y3)
      | ( X7 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X7: real,Y3: real] :
      ( ( X7 != Y3 )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X7),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X7: real,Y3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X7),Y3)
      | ( X7 = Y3 ) ) ).

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,
    ! [X7: complex,Y3: complex] :
      ( ( X7 != Y3 )
      | aa_complex_o(aa_com5808801942499186519plex_o(fequal_complex,X7),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X7: complex,Y3: complex] :
      ( ~ aa_complex_o(aa_com5808801942499186519plex_o(fequal_complex,X7),Y3)
      | ( X7 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X7: set_nat,Y3: set_nat] :
      ( ( X7 != Y3 )
      | aa_set_nat_o(fequal_set_nat(X7),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X7: set_nat,Y3: set_nat] :
      ( ~ aa_set_nat_o(fequal_set_nat(X7),Y3)
      | ( X7 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X7: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( X7 != Y3 )
      | aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X7),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X7: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X7),Y3)
      | ( X7 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X7: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( X7 != Y3 )
      | aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(fequal6010348034681699726nt_int,X7),Y3) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X7: product_prod_int_int,Y3: product_prod_int_int] :
      ( ~ aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(fequal6010348034681699726nt_int,X7),Y3)
      | ( X7 = Y3 ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary)),xa)),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,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

%------------------------------------------------------------------------------