%------------------------------------------------------------------------------
% File     : ITP291_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Intf_Imperative 00092_002426
% 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    : 0096_VEBT_Intf_Imperative_00092_002426 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12113 (4066 unt;1550 typ;   0 def)
%            Number of atoms       : 24946 (9111 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 15921 (2465   ~; 563   |;1214   &)
%                                         (2876 <=>;8803  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   16 (   2 avg)
%            Number of FOOLs       :  973 ( 517 fml; 456 var)
%            Number of X terms     :  999 (   0  []; 722 ite; 277 let)
%            Number of types       :  265 ( 264 usr)
%            Number of type conns  : 1416 ( 918   >; 498   *;   0   +;   0  <<)
%            Number of predicates  :   89 (  86 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1219 (1219 usr; 387 con; 0-4 aty)
%            Number of variables   : 24967 (24222   !; 745   ?;24967   :)
% 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 23:24:49.469
%------------------------------------------------------------------------------
% Could-be-implicit typings (264)
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    fun_num_nat: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
    fun_o_assn: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____3_J,type,
    itself_finite_3: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____2_J,type,
    itself_finite_2: $tType ).

tff(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
    option_rat: $tType ).

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

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

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

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

tff(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
    filter_int: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__Uint32__Ouint32,type,
    uint32: $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 (1286)
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_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: vEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: product_prod_int_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_complex_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_real_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_int_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: vEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: product_prod_int_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: set_VEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_real_real ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_int_int ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_assn_assn ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_real_real ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_int_int ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: set_VEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_Pr958786334691620121nt_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_VEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_VEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: vEBT_VEBT > fun_VE1035329831157012416o_assn ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: vEBT_VEBT > fun_VE9201042017749490615o_assn ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: nat > fun_se292914136812062355o_assn ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: vEBT_VEBT > fun_VE7792148054706007876t_assn ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: vEBT_VEBT > fun_VE7792148054706007876t_assn ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_complex_o > fun_complex_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_Pr3238374608701176162_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_nat > fun_nat_o ).

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

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

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_nat_int ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_nat_real ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: assn > fun_o_fun_o_assn ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: assn > fun_op6483474040057813218t_assn ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: assn > fun_VE2561185598974572172i_assn ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: assn > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_nat_o > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: set_int > fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: set_complex > fun_se823489619503225858plex_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: set_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_nat_o > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_o_assn > fun_assn_fun_o_assn ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_nat_fun_nat_o ).

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: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: int > fun_int_fun_int_o ).

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

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_real > fun_fu6899955116716589935t_real ).

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

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

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_nat_real ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: nat > fun_li7327393198025160363T_VEBT ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: option4927543243414619207at_nat > fun_na3226088584868508064T_VEBT ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: nat > fun_li7327393198025160363T_VEBT ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: nat > fun_li9150060874205195607on_nat ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_na4138050610385705875at_nat ).

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

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

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_real_o ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: real > fun_int_o ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_int_o ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: nat > fun_se8971127763498933158T_assn ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: $o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_int_int ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_int_int ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_int_fun_o_int ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__lq____,type,
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tff(sy_c_ATP_058Lamm__lr____,type,
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tff(sy_c_ATP_058Lamm__ls____,type,
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tff(sy_c_ATP_058Lamm__lt____,type,
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tff(sy_c_ATP_058Lamm__lu____,type,
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tff(sy_c_ATP_058Lamm__lv____,type,
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tff(sy_c_ATP_058Lamm__lw____,type,
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tff(sy_c_ATP_058Lamm__lx____,type,
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tff(sy_c_ATP_058Lamm__ly____,type,
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tff(sy_c_ATP_058Lamm__lz____,type,
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tff(sy_c_ATP_058Lamm__ma____,type,
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tff(sy_c_ATP_058Lamm__mb____,type,
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tff(sy_c_ATP_058Lamm__mc____,type,
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tff(sy_c_ATP_058Lamm__md____,type,
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tff(sy_c_ATP_058Lamm__me____,type,
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tff(sy_c_ATP_058Lamm__mf____,type,
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tff(sy_c_ATP_058Lamm__mg____,type,
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tff(sy_c_ATP_058Lamm__mh____,type,
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tff(sy_c_ATP_058Lamm__mi____,type,
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tff(sy_c_ATP_058Lamm__mj____,type,
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tff(sy_c_ATP_058Lamm__mk____,type,
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tff(sy_c_ATP_058Lamm__ml____,type,
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tff(sy_c_ATP_058Lamm__mm____,type,
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tff(sy_c_ATP_058Lamm__mn____,type,
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tff(sy_c_ATP_058Lamm__mo____,type,
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tff(sy_c_ATP_058Lamm__mp____,type,
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tff(sy_c_ATP_058Lamm__mq____,type,
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tff(sy_c_ATP_058Lamm__mr____,type,
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tff(sy_c_ATP_058Lamm__ms____,type,
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tff(sy_c_ATP_058Lamm__mt____,type,
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tff(sy_c_ATP_058Lamm__mu____,type,
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tff(sy_c_ATP_058Lamm__mv____,type,
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tff(sy_c_ATP_058Lamm__mw____,type,
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tff(sy_c_ATP_058Lamm__mx____,type,
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tff(sy_c_ATP_058Lamm__my____,type,
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tff(sy_c_Assertions_Oentails,type,
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tff(sy_c_Complex_Ocomplex_OComplex,type,
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tff(sy_c_Complex_Ocsqrt,type,
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tff(sy_c_Complex_Oimaginary__unit,type,
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tff(sy_c_Divides_Odivmod__nat,type,
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tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
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tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
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tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
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tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oat__top_001t__Int__Oint,type,
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tff(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
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tff(sy_c_Filter_Ofilterlim_001t__Int__Oint_001t__Nat__Onat,type,
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    inf_inf_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Rat__Orat,type,
    inf_inf_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
    inf_inf_real: real > fun_real_real ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    inf_inf_set_complex: ( set_complex * set_complex ) > set_complex ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Int__Oint_J,type,
    inf_inf_set_int: ( set_int * set_int ) > set_int ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
    inf_inf_set_nat: fun_se1909708154429037836et_nat ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    inf_in2269163501485487111nt_int: ( set_Pr958786334691620121nt_int * set_Pr958786334691620121nt_int ) > set_Pr958786334691620121nt_int ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
    inf_inf_set_real: ( set_real * set_real ) > set_real ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    inf_in473752961466201218T_VEBT: ( set_VEBT_VEBT * set_VEBT_VEBT ) > set_VEBT_VEBT ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    inf_in8039933884210822705plex_o: ( fun_complex_o * fun_complex_o ) > fun_complex_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Int__Oint_M_Eo_J,type,
    inf_inf_fun_int_o: ( fun_int_o * fun_int_o ) > fun_int_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Nat__Onat_M_Eo_J,type,
    inf_inf_fun_nat_o: ( fun_nat_o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    inf_in1782598092406300404_int_o: ( fun_Pr3238374608701176162_int_o * fun_Pr3238374608701176162_int_o ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Real__Oreal_M_Eo_J,type,
    inf_inf_fun_real_o: ( fun_real_o * fun_real_o ) > fun_real_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    inf_in3591633114662893881VEBT_o: ( fun_VEBT_VEBT_o * fun_VEBT_VEBT_o ) > fun_VEBT_VEBT_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Assertions__Oassn,type,
    sup_sup_assn: fun_as3983628580203722482n_assn ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
    sup_su3973961784419623482d_enat: fun_Ex8341929632020906428d_enat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    sup_sup_filter_nat: ( filter_nat * filter_nat ) > filter_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
    sup_sup_int: fun_int_fun_int_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Assertions__Oassn_J,type,
    sup_sup_option_assn: ( option_assn * option_assn ) > option_assn ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Extended____Nat__Oenat_J,type,
    sup_su2190601396275437440d_enat: ( option_Extended_enat * option_Extended_enat ) > option_Extended_enat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Int__Oint_J,type,
    sup_sup_option_int: ( option_int * option_int ) > option_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Nat__Onat_J,type,
    sup_sup_option_nat: ( option_nat * option_nat ) > option_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_su3598758113090618626et_nat: ( option_set_nat * option_set_nat ) > option_set_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
    sup_sup_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
    sup_sup_real: real > fun_real_real ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    sup_sup_set_complex: ( set_complex * set_complex ) > set_complex ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
    sup_sup_set_int: ( set_int * set_int ) > set_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: fun_se1909708154429037836et_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Num__Onum_J,type,
    sup_sup_set_num: ( set_num * set_num ) > set_num ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    sup_su6024340866399070445nt_int: ( set_Pr958786334691620121nt_int * set_Pr958786334691620121nt_int ) > set_Pr958786334691620121nt_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
    sup_sup_set_rat: ( set_rat * set_rat ) > set_rat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
    sup_sup_set_real: ( set_real * set_real ) > set_real ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    sup_su6272177626956685416T_VEBT: ( set_VEBT_VEBT * set_VEBT_VEBT ) > set_VEBT_VEBT ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    sup_su37214000559163671plex_o: ( fun_complex_o * fun_complex_o ) > fun_complex_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Int__Oint_M_Eo_J,type,
    sup_sup_fun_int_o: ( fun_int_o * fun_int_o ) > fun_int_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Nat__Onat_M_Eo_J,type,
    sup_sup_fun_nat_o: ( fun_nat_o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    sup_su4047618219528032398_int_o: ( fun_Pr3238374608701176162_int_o * fun_Pr3238374608701176162_int_o ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Real__Oreal_M_Eo_J,type,
    sup_sup_fun_real_o: ( fun_real_o * fun_real_o ) > fun_real_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    sup_su3075289160232154067VEBT_o: ( fun_VEBT_VEBT_o * fun_VEBT_VEBT_o ) > fun_VEBT_VEBT_o ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
    lattic8263393255366662781ax_int: set_int > int ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Code____Numeral__Ointeger,type,
    lattic1063845414844153500nteger: set_Code_integer > code_integer ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Int__Oint,type,
    lattic8718645017227715691in_int: set_int > int ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
    lattic8721135487736765967in_nat: set_nat > nat ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Num__Onum,type,
    lattic5278467273892544601in_num: set_num > num ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Rat__Orat,type,
    lattic8086005427650270231in_rat: set_rat > rat ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Real__Oreal,type,
    lattic3629708407755379051n_real: set_real > real ).

tff(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Int__Oint,type,
    least_4859182151741483524sb_int: fun_int_o ).

tff(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
    foldr_nat_nat: ( fun_nat_fun_nat_nat * list_nat ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
    foldr_real_real: ( fun_re7681423839302404990l_real * list_real ) > fun_real_real ).

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

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

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

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

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    map_VEBT_VEBT_nat: ( fun_VEBT_VEBT_nat * list_VEBT_VEBT ) > list_nat ).

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

tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

tff(sy_c_List_Olist__update_001t__VEBT____BuildupMemImp__OVEBTi,type,
    list_u6098035379799741383_VEBTi: ( list_VEBT_VEBTi * nat * vEBT_VEBTi ) > list_VEBT_VEBTi ).

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

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

tff(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    nth_VEBT_VEBTi: ( list_VEBT_VEBTi * nat ) > vEBT_VEBTi ).

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

tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
    replicate_VEBT_VEBT: ( nat * vEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: ( $o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num * nat ) > option_num ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: fun_num_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
    neg_nu8804712462038260780nteger: code_integer > code_integer ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: complex > complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: int > int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: complex > complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: int > int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
    case_num_option_num: ( option_num * fun_num_option_num * fun_num_option_num * num ) > option_num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: fun_num_Code_integer ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
    numera6690914467698888265omplex: num > complex ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: fun_num_rat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: fun_num_real ).

tff(sy_c_Num_Opow,type,
    pow: num > fun_num_num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Assertions__Oassn,type,
    some_assn: assn > option_assn ).

tff(sy_c_Option_Ooption_OSome_001t__Code____Numeral__Ointeger,type,
    some_Code_integer: code_integer > option_Code_integer ).

tff(sy_c_Option_Ooption_OSome_001t__Extended____Nat__Oenat,type,
    some_Extended_enat: extended_enat > option_Extended_enat ).

tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: num > option_num ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
    some_rat: rat > option_rat ).

tff(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
    some_real: real > option_real ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
    some_set_nat: set_nat > option_set_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
    case_option_int_num: ( int * fun_num_int * option_num ) > int ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
    case_option_num_num: ( num * fun_num_num * option_num ) > num ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o1383228350324149268at_nat: ( option_nat * fun_Pr2458574587993444404on_nat * option4927543243414619207at_nat ) > option_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    case_o6005452278849405969um_num: ( option_num * fun_num_option_num * option_num ) > option_num ).

tff(sy_c_Option_Ooption_Ocase__option_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o2442805151034396888at_nat: ( vEBT_VEBT * fun_Pr4536201527729328438T_VEBT * option4927543243414619207at_nat ) > vEBT_VEBT ).

tff(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
    the_nat: option_nat > nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
    bot_bot_assn: assn ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    bot_bot_filter_nat: filter_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Option__Ooption_It__Nat__Onat_J,type,
    bot_bot_option_nat: option_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Option__Ooption_It__Num__Onum_J,type,
    bot_bot_option_num: option_num ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    bot_bo3990330152332043303nteger: set_Code_integer ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    bot_bo8750099748576881071plex_o: fun_complex_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Int__Oint_M_Eo_J,type,
    bot_bot_fun_int_o: fun_int_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Nat__Onat_M_Eo_J,type,
    bot_bot_fun_nat_o: fun_nat_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Real__Oreal_M_Eo_J,type,
    bot_bot_fun_real_o: fun_real_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    bot_bo6217218057720408379VEBT_o: fun_VEBT_VEBT_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Assertions__Oassn,type,
    ord_less_assn: ( assn * assn ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > fun_Extended_enat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_less_filter_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Code____Numeral__Ointeger_J,type,
    ord_le7113747843092208513nteger: ( option_Code_integer * option_Code_integer ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_less_option_int: ( option_int * option_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: ( option_nat * option_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_less_option_num: ( option_num * option_num ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_less_option_rat: ( option_rat * option_rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
    ord_less_option_real: ( option_real * option_real ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > fun_set_complex_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: set_int > fun_set_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > fun_set_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: set_num > fun_set_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: set_rat > fun_set_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_set_real: set_real > fun_set_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: ( set_set_nat * 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__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__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_le3534172602983796987VEBT_o: ( fun_VEBT_VEBT_o * fun_VEBT_VEBT_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Assertions__Oassn,type,
    ord_less_eq_assn: ( assn * assn ) > $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 > fun_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__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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__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__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__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__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__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__Real__Oreal_M_Eo_J,type,
    ord_le3605727316730466277real_o: ( fun_real_o * fun_real_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_le88303844413636359VEBT_o: ( fun_VEBT_VEBT_o * fun_VEBT_VEBT_o ) > $o ).

tff(sy_c_Orderings_Oord__class_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_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: fun_Ex8341929632020906428d_enat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
    ord_min_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

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__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Power_Opower__class_Opower_001t__Assertions__Oassn,type,
    power_power_assn: assn > fun_nat_assn ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > fun_nat_complex ).

tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > fun_nat_rat ).

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_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: int > fun_in4211310926729683942nt_int ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > fun_na1476630412504255114at_nat ).

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

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

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_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( fun_Co2470855966376689427nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: fun_int_fun_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: ( fun_int_fun_int_int * product_prod_int_int ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( fun_in1696471567231710611nt_int * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
    produc2484365769952853102on_nat: fun_na7846659920737657402on_nat > fun_Pr2458574587993444404on_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( fun_na4138050610385705875at_nat * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    produc478579273971653890on_num: ( fun_na7595719129879163854on_num * product_prod_nat_num ) > option_num ).

tff(sy_c_Pure_Otype_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_N8448461349408098053l_num1: itself8794530163899892676l_num1 ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V5970128139526366754l_real: fun_real_real > $o ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: complex > real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: real > complex ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: real > fun_real_real ).

tff(sy_c_Refine__Imp__Hol_Orefines_001_Eo,type,
    refine_Imp_refines_o: ( heap_Time_Heap_o * heap_Time_Heap_o ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_I_Eo_J,type,
    refine5896690332125372649list_o: ( heap_T844314716496656296list_o * heap_T844314716496656296list_o ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    refine1935026298455697829on_nat: ( heap_T5317711798761887292on_nat * heap_T5317711798761887292on_nat ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    refine3700189196150522554_VEBTi: ( heap_T4980287057938770641_VEBTi * heap_T4980287057938770641_VEBTi ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__Nat__Onat,type,
    refine1365783493865988805es_nat: ( heap_Time_Heap_nat * heap_Time_Heap_nat ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__Option__Ooption_It__Nat__Onat_J,type,
    refine7594492741263601813on_nat: ( heap_T2636463487746394924on_nat * heap_T2636463487746394924on_nat ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__VEBT____BuildupMemImp__OVEBTi,type,
    refine5565527176597971370_VEBTi: ( heap_T8145700208782473153_VEBTi * heap_T8145700208782473153_VEBTi ) > $o ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: ( complex * complex ) > complex ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > fun_real_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: fun_Co2081214437782444048eger_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: complex > fun_complex_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_real_fun_real_o ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_o_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
    suminf_complex: fun_nat_complex > complex ).

tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
    suminf_int: fun_nat_int > int ).

tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
    suminf_nat: fun_nat_nat > nat ).

tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: fun_nat_real > real ).

tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
    summable_complex: fun_nat_complex > $o ).

tff(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: fun_nat_int > $o ).

tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: fun_nat_nat > $o ).

tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: fun_nat_real > $o ).

tff(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: fun_nat_real > fun_real_o ).

tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: fun_complex_o > set_complex ).

tff(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: fun_int_o > set_int ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: fun_nat_o > set_nat ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: fun_Pr3238374608701176162_int_o > set_Pr958786334691620121nt_int ).

tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: fun_real_o > set_real ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    collect_set_complex: fun_set_complex_o > set_set_complex ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: fun_set_int_o > set_set_int ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: fun_set_nat_o > set_set_nat ).

tff(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
    collect_VEBT_VEBT: fun_VEBT_VEBT_o > set_VEBT_VEBT ).

tff(sy_c_Set_Oimage_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    image_4470545334726330049nteger: ( fun_Co8499238673658658103nteger * set_Code_integer ) > set_Code_integer ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( fun_int_int * set_int ) > set_int ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( fun_int_nat * set_int ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
    image_nat_int: ( fun_nat_int * set_nat ) > set_int ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( fun_real_real * set_real ) > set_real ).

tff(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    image_VEBT_VEBT_nat: ( fun_VEBT_VEBT_nat * set_VEBT_VEBT ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Code____Numeral__Ointeger,type,
    insert_Code_integer: ( code_integer * set_Code_integer ) > set_Code_integer ).

tff(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
    insert_complex: ( complex * set_complex ) > set_complex ).

tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: ( int * set_int ) > set_int ).

tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: ( nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Num__Onum,type,
    insert_num: ( num * set_num ) > set_num ).

tff(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    insert5033312907999012233nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > set_Pr958786334691620121nt_int ).

tff(sy_c_Set_Oinsert_001t__Rat__Orat,type,
    insert_rat: ( rat * set_rat ) > set_rat ).

tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: ( real * set_real ) > set_real ).

tff(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
    insert_VEBT_VEBT: ( vEBT_VEBT * set_VEBT_VEBT ) > set_VEBT_VEBT ).

tff(sy_c_Set_Othe__elem_001t__Int__Oint,type,
    the_elem_int: set_int > int ).

tff(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: set_nat > nat ).

tff(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
    the_elem_real: set_real > real ).

tff(sy_c_Set_Othe__elem_001t__VEBT____Definitions__OVEBT,type,
    the_elem_VEBT_VEBT: set_VEBT_VEBT > vEBT_VEBT ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Code____Numeral__Ointeger,type,
    set_or189985376899183464nteger: ( code_integer * code_integer ) > set_Code_integer ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
    set_or7049704709247886629st_num: ( num * num ) > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
    set_or633870826150836451st_rat: ( rat * rat ) > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
    set_or1222579329274155063t_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: ( set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Code____Numeral__Ointeger,type,
    set_or8404916559141939852nteger: ( code_integer * code_integer ) > set_Code_integer ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
    set_or1633881224788618240n_real: ( real * real ) > set_real ).

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_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
    set_or5984915006950818249n_real: real > set_real ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__divide_001t__Int__Oint,type,
    signed6714573509424544716de_int: fun_int_fun_int_int ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
    signed6292675348222524329lo_int: fun_int_fun_int_int ).

tff(sy_c_Time__Reasoning_OTBOUND_001_Eo,type,
    time_TBOUND_o: heap_Time_Heap_o > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_I_Eo_J,type,
    time_TBOUND_list_o: heap_T844314716496656296list_o > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Nat__Onat_J,type,
    time_TBOUND_list_nat: heap_T290393402774840812st_nat > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    time_T3808005469503390304on_nat: heap_T5317711798761887292on_nat > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T8149879359713347829_VEBTi: heap_T4980287057938770641_VEBTi > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Nat__Onat,type,
    time_TBOUND_nat: heap_Time_Heap_nat > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_T8353473612707095248on_nat: heap_T2636463487746394924on_nat > fun_nat_o ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_T5737551269749752165_VEBTi: heap_T8145700208782473153_VEBTi > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ohtt_001_Eo,type,
    time_htt_o: ( assn * heap_Time_Heap_o * fun_o_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__Nat__Onat,type,
    time_htt_nat: ( assn * heap_Time_Heap_nat * fun_nat_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_htt_option_nat: ( assn * heap_T2636463487746394924on_nat * fun_option_nat_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_htt_VEBT_VEBTi: ( assn * heap_T8145700208782473153_VEBTi * fun_VEBT_VEBTi_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t3534373299052942712_VEBTi: ( heap_T4980287057938770641_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: ( heap_T8145700208782473153_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

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

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: ( real * set_real ) > filter_real ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: fun_nat_real > $o ).

tff(sy_c_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: complex > complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
    diffs_real: fun_nat_real > fun_nat_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: 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: 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: complex > complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: complex > complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

tff(sy_c_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: 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: 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,type,
    vEBT_T_m_i_n_t: 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__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: ( nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: ( nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001_Eo,type,
    vEBT_V2326993469660664182atei_o: ( nat * heap_Time_Heap_o ) > heap_T844314716496656296list_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: ( nat * heap_Time_Heap_nat ) > heap_T290393402774840812st_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_V792416675989592002on_nat: ( nat * heap_T2636463487746394924on_nat ) > heap_T5317711798761887292on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: ( nat * heap_T8145700208782473153_VEBTi ) > heap_T4980287057938770641_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    vEBT_c6104975476656191286Heap_o: ( fun_op1763656958002597022Heap_o * fun_o_1207901889202581752Heap_o * vEBT_VEBTi ) > heap_Time_Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    vEBT_c6250501799366334488on_nat: ( fun_op5783775701855621896on_nat * fun_o_4049874799390722306on_nat * vEBT_VEBTi ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( fun_op5127434927081926045_VEBTi * fun_o_6972675622308511895_VEBTi * vEBT_VEBTi ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun_VE905575822219544865i_assn ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: ( vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: ( vEBT_VEBTi * nat ) > heap_Time_Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > vEBT_VEBT ).

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

tff(sy_c_VEBT__Definitions_OVEBT__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_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__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
    vEBT_V1365221501068881998eletei: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
    vEBT_vebt_deletei: ( vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

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__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
    vEBT_VEBT_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
    vEBT_Intf_vebt_assn: ( nat * set_nat ) > fun_VEBT_VEBTi_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: ( set_nat * fun_VE905575822219544865i_assn * list_VEBT_VEBT * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( fun_VE905575822219544865i_assn * list_VEBT_VEBT * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_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: ( option_nat * option_nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option_nat * option_nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option_nat * option_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_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__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

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__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: vEBT_VEBT > real ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: ( vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: ( vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option_nat ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Uniqueness_OVEBT__internal_OperInsTrans,type,
    vEBT_V6289311342943941716sTrans: vEBT_VEBT > fun_VEBT_VEBT_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__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr8877776288780692068_nat_o * produc9072475918466114483BT_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_aa_001_Eo_001_Eo,type,
    aa_o_o: ( fun_o_o * $o ) > $o ).

tff(sy_c_aa_001_Eo_001t__Assertions__Oassn,type,
    aa_o_assn: ( fun_o_assn * $o ) > assn ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    aa_o_H7399033255965945470Heap_o: ( fun_o_8325599348076409421Heap_o * $o ) > heap_Time_Heap_o ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    aa_o_H4215342640551218896on_nat: ( fun_o_1929998731553720919on_nat * $o ) > heap_T2636463487746394924on_nat ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    aa_o_H7577119287286566245_VEBTi: ( fun_o_2238389867833408620_VEBTi * $o ) > heap_T8145700208782473153_VEBTi ).

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__fun_I_Eo_M_Eo_J,type,
    aa_o_fun_o_o: ( fun_o_fun_o_o * $o ) > fun_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
    aa_o_fun_o_assn: ( fun_o_fun_o_assn * $o ) > fun_o_assn ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    aa_o_f869145629572826609Heap_o: ( fun_o_1207901889202581752Heap_o * $o ) > fun_o_8325599348076409421Heap_o ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    aa_o_f5606600903069576627on_nat: ( fun_o_4049874799390722306on_nat * $o ) > fun_o_1929998731553720919on_nat ).

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tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun1113820866546176217t_real: ( fun_fu5589980185390252312t_real * fun_na6724076748692969250l_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
    aa_fun2567993205388704171l_real: ( fun_fu3612569900928539882l_real * fun_na6724076748692969250l_real ) > fun_re4112884445705685835l_real ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun5349570537165229698_int_o: ( fun_fu3068557634114553409_int_o * fun_Pr3238374608701176162_int_o ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_M_Eo_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun5868358341814070780real_o: ( fun_fu9050450510664235267real_o * fun_real_o ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun2895569901277098738real_o: ( fun_fu3022329567491847689real_o * fun_real_real ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_fun6946558556880649704l_real: ( fun_fu3077872570260440615l_real * fun_real_real ) > fun_real_real ).

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_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_o > real ).

tff(sy_c_fequal_001_Eo,type,
    fequal_o: fun_o_fun_o_o ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: complex > fun_complex_o ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_o ).

tff(sy_c_fequal_001t__Num__Onum,type,
    fequal_num: fun_num_fun_num_o ).

tff(sy_c_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    fequal6010348034681699726nt_int: product_prod_int_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_real_fun_real_o ).

tff(sy_c_fequal_001t__VEBT____Definitions__OVEBT,type,
    fequal_VEBT_VEBT: vEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > fun_set_complex_o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: int > fun_set_int_o ).

tff(sy_c_member_001t__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__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > fun_set_rat_o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: real > fun_set_real_o ).

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

tff(sy_v_n,type,
    n: nat ).

tff(sy_v_s,type,
    s: set_nat ).

tff(sy_v_ti,type,
    ti: vEBT_VEBTi ).

tff(sy_v_x,type,
    x: nat ).

% Relevant facts (10544)
tff(fact_0_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_1_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Na)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: 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)),Na)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: 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)),Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: 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)),Na)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_8_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_9_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_10_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_11_perIT__concat,axiom,
    ! [Sa: vEBT_VEBT,T: vEBT_VEBT,U: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(Sa),T)
     => ( aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(T),U)
       => aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(Sa),U) ) ) ).

% perIT_concat
tff(fact_12_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: 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,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_13_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: 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,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_14_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: 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,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_15_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_16_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: 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,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_17_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_18_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_19_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_20_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_21_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_22_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),Na) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_23_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xa)),Na) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_24_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_25_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_26_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_27_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_28_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Na: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_29_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Na: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xa)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_30_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_31_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_32_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_33_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_34_ceiling__numeral__power,axiom,
    ! [Xa: num,Na: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) ).

% ceiling_numeral_power
tff(fact_35_ceiling__numeral__power,axiom,
    ! [Xa: num,Na: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) ).

% ceiling_numeral_power
tff(fact_36_ceiling__add__numeral,axiom,
    ! [Xa: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_37_ceiling__add__numeral,axiom,
    ! [Xa: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_38_numeral__less__ceiling,axiom,
    ! [V: num,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,V)),Xa) ) ).

% numeral_less_ceiling
tff(fact_39_numeral__less__ceiling,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),Xa) ) ).

% numeral_less_ceiling
tff(fact_40_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,M)),Na) ).

% semiring_1_class.of_nat_power
tff(fact_41_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),Na) ).

% semiring_1_class.of_nat_power
tff(fact_42_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),Na) ).

% semiring_1_class.of_nat_power
tff(fact_43_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),Na) ).

% semiring_1_class.of_nat_power
tff(fact_44_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),Na) ).

% semiring_1_class.of_nat_power
tff(fact_45_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),Na) ).

% semiring_1_class.of_nat_power
tff(fact_46_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_47_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_48_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_49_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) = aa_nat_rat(semiri681578069525770553at_rat,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_50_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_51_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_52_ceiling__of__nat,axiom,
    ! [Na: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% ceiling_of_nat
tff(fact_53_ceiling__of__nat,axiom,
    ! [Na: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% ceiling_of_nat
tff(fact_54_power__add__numeral2,axiom,
    ! [A: complex,M: num,Na: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Na))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na)))),B) ).

% power_add_numeral2
tff(fact_55_power__add__numeral2,axiom,
    ! [A: code_integer,M: num,Na: num,B: code_integer] : times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M)),times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,Na)),B)) = times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na))),B) ).

% power_add_numeral2
tff(fact_56_power__add__numeral2,axiom,
    ! [A: real,M: num,Na: num,B: real] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Na))),B)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na)))),B) ).

% power_add_numeral2
tff(fact_57_power__add__numeral2,axiom,
    ! [A: rat,M: num,Na: num,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Na))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na)))),B) ).

% power_add_numeral2
tff(fact_58_power__add__numeral2,axiom,
    ! [A: nat,M: num,Na: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Na))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na)))),B) ).

% power_add_numeral2
tff(fact_59_power__add__numeral2,axiom,
    ! [A: int,M: num,Na: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Na))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na)))),B) ).

% power_add_numeral2
tff(fact_60_power__add__numeral,axiom,
    ! [A: complex,M: num,Na: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Na))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na))) ).

% power_add_numeral
tff(fact_61_power__add__numeral,axiom,
    ! [A: code_integer,M: num,Na: num] : times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,Na))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na))) ).

% power_add_numeral
tff(fact_62_power__add__numeral,axiom,
    ! [A: real,M: num,Na: num] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Na))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na))) ).

% power_add_numeral
tff(fact_63_power__add__numeral,axiom,
    ! [A: rat,M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Na))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na))) ).

% power_add_numeral
tff(fact_64_power__add__numeral,axiom,
    ! [A: nat,M: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na))) ).

% power_add_numeral
tff(fact_65_power__add__numeral,axiom,
    ! [A: int,M: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Na))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na))) ).

% power_add_numeral
tff(fact_66_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xa) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_67_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xa) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_68_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xa) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_69_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Xa) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_70_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Xa) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_71_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xa) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_72_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_73_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_74_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_num_real(numeral_numeral_real,W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_75_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_76_power__commuting__commutes,axiom,
    ! [Xa: complex,Y: complex,Na: nat] :
      ( ( aa_complex_complex(times_times_complex(Xa),Y) = aa_complex_complex(times_times_complex(Y),Xa) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xa),Na)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(Xa),Na)) ) ) ).

% power_commuting_commutes
tff(fact_77_power__commuting__commutes,axiom,
    ! [Xa: code_integer,Y: code_integer,Na: nat] :
      ( ( times_3573771949741848930nteger(Xa,Y) = times_3573771949741848930nteger(Y,Xa) )
     => ( times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Na),Y) = times_3573771949741848930nteger(Y,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Na)) ) ) ).

% power_commuting_commutes
tff(fact_78_power__commuting__commutes,axiom,
    ! [Xa: real,Y: real,Na: nat] :
      ( ( aa_real_real(times_times_real(Xa),Y) = aa_real_real(times_times_real(Y),Xa) )
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)),Y) = aa_real_real(times_times_real(Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) ) ) ).

% power_commuting_commutes
tff(fact_79_power__commuting__commutes,axiom,
    ! [Xa: rat,Y: rat,Na: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xa) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xa),Na)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(Xa),Na)) ) ) ).

% power_commuting_commutes
tff(fact_80_power__commuting__commutes,axiom,
    ! [Xa: nat,Y: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),Xa) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na)) ) ) ).

% power_commuting_commutes
tff(fact_81_power__commuting__commutes,axiom,
    ! [Xa: int,Y: int,Na: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xa) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xa),Na)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(Xa),Na)) ) ) ).

% power_commuting_commutes
tff(fact_82_power__mult__distrib,axiom,
    ! [A: complex,B: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),Na) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Na)),aa_nat_complex(power_power_complex(B),Na)) ).

% power_mult_distrib
tff(fact_83_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(times_3573771949741848930nteger(A,B)),Na) = times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na)) ).

% power_mult_distrib
tff(fact_84_power__mult__distrib,axiom,
    ! [A: real,B: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(A),B)),Na) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ).

% power_mult_distrib
tff(fact_85_power__mult__distrib,axiom,
    ! [A: rat,B: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),Na) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ).

% power_mult_distrib
tff(fact_86_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ).

% power_mult_distrib
tff(fact_87_power__mult__distrib,axiom,
    ! [A: int,B: int,Na: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ).

% power_mult_distrib
tff(fact_88_power__commutes,axiom,
    ! [A: complex,Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Na)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Na)) ).

% power_commutes
tff(fact_89_power__commutes,axiom,
    ! [A: code_integer,Na: nat] : times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na),A) = times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ).

% power_commutes
tff(fact_90_power__commutes,axiom,
    ! [A: real,Na: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),A) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ).

% power_commutes
tff(fact_91_power__commutes,axiom,
    ! [A: rat,Na: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Na)),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na)) ).

% power_commutes
tff(fact_92_power__commutes,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ).

% power_commutes
tff(fact_93_power__commutes,axiom,
    ! [A: int,Na: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Na)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na)) ).

% power_commutes
tff(fact_94_reals__Archimedean2,axiom,
    ! [Xa: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_95_reals__Archimedean2,axiom,
    ! [Xa: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_96_power__mult,axiom,
    ! [A: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),Na) ).

% power_mult
tff(fact_97_power__mult,axiom,
    ! [A: int,M: nat,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),M)),Na) ).

% power_mult
tff(fact_98_power__mult,axiom,
    ! [A: real,M: nat,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),Na) ).

% power_mult
tff(fact_99_power__mult,axiom,
    ! [A: complex,M: nat,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),M)),Na) ).

% power_mult
tff(fact_100_power__mult,axiom,
    ! [A: code_integer,M: nat,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),Na) ).

% power_mult
tff(fact_101_power__mult,axiom,
    ! [A: rat,M: nat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),M)),Na) ).

% power_mult
tff(fact_102_ceiling__less__cancel,axiom,
    ! [Xa: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_real_int(archim7802044766580827645g_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% ceiling_less_cancel
tff(fact_103_ceiling__less__cancel,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_rat_int(archim2889992004027027881ng_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ).

% ceiling_less_cancel
tff(fact_104_power__add,axiom,
    ! [A: complex,M: nat,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),Na)) ).

% power_add
tff(fact_105_power__add,axiom,
    ! [A: code_integer,M: nat,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ).

% power_add
tff(fact_106_power__add,axiom,
    ! [A: real,M: nat,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ).

% power_add
tff(fact_107_power__add,axiom,
    ! [A: rat,M: nat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na)) ).

% power_add
tff(fact_108_power__add,axiom,
    ! [A: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ).

% power_add
tff(fact_109_power__add,axiom,
    ! [A: int,M: nat,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na)) ).

% power_add
tff(fact_110_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(w),w) ) ).

% power_numeral_even
tff(fact_111_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W)),
        times_3573771949741848930nteger(w,w) ) ).

% power_numeral_even
tff(fact_112_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_113_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(aa_rat_fun_rat_rat(times_times_rat,w),w) ) ).

% power_numeral_even
tff(fact_114_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_115_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_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_116_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_117_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W)),
        times_3573771949741848930nteger(times_3573771949741848930nteger(Z,w),w) ) ).

% power_numeral_odd
tff(fact_118_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_119_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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_120_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_121_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_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_122_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(times_times_complex(A),A) ).

% power2_eq_square
tff(fact_123_power2__eq__square,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = times_3573771949741848930nteger(A,A) ).

% power2_eq_square
tff(fact_124_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(A),A) ).

% power2_eq_square
tff(fact_125_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% power2_eq_square
tff(fact_126_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_127_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% power2_eq_square
tff(fact_128_power4__eq__xxxx,axiom,
    ! [Xa: complex] : aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_129_power4__eq__xxxx,axiom,
    ! [Xa: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = times_3573771949741848930nteger(times_3573771949741848930nteger(times_3573771949741848930nteger(Xa,Xa),Xa),Xa) ).

% power4_eq_xxxx
tff(fact_130_power4__eq__xxxx,axiom,
    ! [Xa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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(Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_131_power4__eq__xxxx,axiom,
    ! [Xa: rat] : aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_132_power4__eq__xxxx,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),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,Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_133_power4__eq__xxxx,axiom,
    ! [Xa: int] : aa_nat_int(power_power_int(Xa),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,Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_134_power3__eq__cube,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),A)),A) ).

% power3_eq_cube
tff(fact_135_power3__eq__cube,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = times_3573771949741848930nteger(times_3573771949741848930nteger(A,A),A) ).

% power3_eq_cube
tff(fact_136_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(times_times_real(aa_real_real(times_times_real(A),A)),A) ).

% power3_eq_cube
tff(fact_137_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),A) ).

% power3_eq_cube
tff(fact_138_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).

% power3_eq_cube
tff(fact_139_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).

% power3_eq_cube
tff(fact_140_power__even__eq,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_141_power__even__eq,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_142_power__even__eq,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_143_power__even__eq,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_144_power__even__eq,axiom,
    ! [A: code_integer,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_145_power__even__eq,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_146_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_147_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_148_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_149_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_150_power2__sum,axiom,
    ! [Xa: complex,Y: complex] : aa_nat_complex(power_power_complex(plus_plus_complex(Xa,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_plus_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_sum
tff(fact_151_power2__sum,axiom,
    ! [Xa: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_sum
tff(fact_152_power2__sum,axiom,
    ! [Xa: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_sum
tff(fact_153_power2__sum,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_sum
tff(fact_154_power2__sum,axiom,
    ! [Xa: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_sum
tff(fact_155_power2__sum,axiom,
    ! [Xa: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(plus_p5714425477246183910nteger(Xa,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_3573771949741848930nteger(times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),Xa),Y)) ).

% power2_sum
tff(fact_156_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_157_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_158_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_159_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_160_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% less_log2_of_power
tff(fact_161_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_162_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_mult
tff(fact_163_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_mult
tff(fact_164_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_mult
tff(fact_165_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_mult
tff(fact_166_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_mult
tff(fact_167_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Na)) ).

% of_nat_mult
tff(fact_168_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_add
tff(fact_169_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_add
tff(fact_170_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_add
tff(fact_171_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_add
tff(fact_172_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_add
tff(fact_173_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M),aa_nat_complex(semiri8010041392384452111omplex,Na)) ).

% of_nat_add
tff(fact_174_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_175_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_176_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_177_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_178_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_179_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_real(numeral_numeral_real,Na) ).

% of_nat_numeral
tff(fact_180_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% of_nat_numeral
tff(fact_181_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,Na) ).

% of_nat_numeral
tff(fact_182_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_rat(numeral_numeral_rat,Na) ).

% of_nat_numeral
tff(fact_183_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% of_nat_numeral
tff(fact_184_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Na)) = numera6690914467698888265omplex(Na) ).

% of_nat_numeral
tff(fact_185_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( aa_set_real_o(member_real(A),collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_186_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),collect_VEBT_VEBT(P))
    <=> aa_VEBT_VEBT_o(P,A) ) ).

% mem_Collect_eq
tff(fact_187_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_188_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_189_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(A),collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_190_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(A,collec213857154873943460nt_int(P))
    <=> aa_Pro4592068669816069595_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_191_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_192_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_193_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_194_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_195_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_196_Collect__mem__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_197_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(P,X)
        <=> aa_nat_o(Q,X) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_198_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( aa_int_o(P,X)
        <=> aa_int_o(Q,X) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_199_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X: complex] :
          ( aa_complex_o(P,X)
        <=> aa_complex_o(Q,X) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_200_Collect__cong,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ! [X: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X)
        <=> aa_Pro4592068669816069595_int_o(Q,X) )
     => ( collec213857154873943460nt_int(P) = collec213857154873943460nt_int(Q) ) ) ).

% Collect_cong
tff(fact_201_semiring__norm_I3_J,axiom,
    ! [Na: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit0,Na)) = aa_num_num(bit1,Na) ).

% semiring_norm(3)
tff(fact_202_semiring__norm_I4_J,axiom,
    ! [Na: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit1,Na)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Na),one)) ).

% semiring_norm(4)
tff(fact_203_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_204_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_real(numeral_numeral_real,M) = aa_num_real(numeral_numeral_real,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_205_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_rat(numeral_numeral_rat,M) = aa_num_rat(numeral_numeral_rat,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_206_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_207_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_208_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_Code_integer(numera6620942414471956472nteger,M) = aa_num_Code_integer(numera6620942414471956472nteger,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_209_semiring__norm_I13_J,axiom,
    ! [M: num,Na: num] : aa_num_num(times_times_num(aa_num_num(bit0,M)),aa_num_num(bit0,Na)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(times_times_num(M),Na))) ).

% semiring_norm(13)
tff(fact_210_semiring__norm_I87_J,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(87)
tff(fact_211_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(78)
tff(fact_212_semiring__norm_I12_J,axiom,
    ! [Na: num] : aa_num_num(times_times_num(one),Na) = Na ).

% semiring_norm(12)
tff(fact_213_semiring__norm_I11_J,axiom,
    ! [M: num] : aa_num_num(times_times_num(M),one) = M ).

% semiring_norm(11)
tff(fact_214_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,M),one) ).

% semiring_norm(75)
tff(fact_215_semiring__norm_I90_J,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(90)
tff(fact_216_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(80)
tff(fact_217_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_218_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_219_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_220_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_221_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_222_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_223_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_224_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_225_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_226_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_227_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_228_num__double,axiom,
    ! [Na: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),Na) = aa_num_num(bit0,Na) ).

% num_double
tff(fact_229_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_230_semiring__norm_I83_J,axiom,
    ! [Na: num] : one != aa_num_num(bit0,Na) ).

% semiring_norm(83)
tff(fact_231_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Na)) ).

% semiring_norm(76)
tff(fact_232_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Na)) = numera6690914467698888265omplex(aa_num_num(times_times_num(M),Na)) ).

% numeral_times_numeral
tff(fact_233_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),Na)) ).

% numeral_times_numeral
tff(fact_234_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),Na)) ).

% numeral_times_numeral
tff(fact_235_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Na)) ).

% numeral_times_numeral
tff(fact_236_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),Na)) ).

% numeral_times_numeral
tff(fact_237_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),Na)) ).

% numeral_times_numeral
tff(fact_238_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_239_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(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_240_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_241_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(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_242_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(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_243_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: code_integer] : times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,V),times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,W),Z)) = times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(V),W)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_244_semiring__norm_I15_J,axiom,
    ! [M: num,Na: num] : aa_num_num(times_times_num(aa_num_num(bit1,M)),aa_num_num(bit0,Na)) = aa_num_num(bit0,aa_num_num(times_times_num(aa_num_num(bit1,M)),Na)) ).

% semiring_norm(15)
tff(fact_245_semiring__norm_I14_J,axiom,
    ! [M: num,Na: num] : aa_num_num(times_times_num(aa_num_num(bit0,M)),aa_num_num(bit1,Na)) = aa_num_num(bit0,aa_num_num(times_times_num(M),aa_num_num(bit1,Na))) ).

% semiring_norm(14)
tff(fact_246_semiring__norm_I89_J,axiom,
    ! [M: num,Na: num] : aa_num_num(bit1,M) != aa_num_num(bit0,Na) ).

% semiring_norm(89)
tff(fact_247_semiring__norm_I88_J,axiom,
    ! [M: num,Na: num] : aa_num_num(bit0,M) != aa_num_num(bit1,Na) ).

% semiring_norm(88)
tff(fact_248_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(81)
tff(fact_249_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_250_semiring__norm_I84_J,axiom,
    ! [Na: num] : one != aa_num_num(bit1,Na) ).

% semiring_norm(84)
tff(fact_251_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Na)) ).

% semiring_norm(77)
tff(fact_252_power__mult__numeral,axiom,
    ! [A: nat,M: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Na))) ).

% power_mult_numeral
tff(fact_253_power__mult__numeral,axiom,
    ! [A: int,M: num,Na: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Na)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Na))) ).

% power_mult_numeral
tff(fact_254_power__mult__numeral,axiom,
    ! [A: real,M: num,Na: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Na))) ).

% power_mult_numeral
tff(fact_255_power__mult__numeral,axiom,
    ! [A: complex,M: num,Na: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Na)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Na))) ).

% power_mult_numeral
tff(fact_256_power__mult__numeral,axiom,
    ! [A: code_integer,M: num,Na: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Na))) ).

% power_mult_numeral
tff(fact_257_power__mult__numeral,axiom,
    ! [A: rat,M: num,Na: num] : aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Na)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Na))) ).

% power_mult_numeral
tff(fact_258_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_259_nat__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,Na)) = Na ).

% nat_int
tff(fact_260_semiring__norm_I6_J,axiom,
    ! [M: num,Na: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),aa_num_num(bit0,Na)) = aa_num_num(bit0,aa_num_num(plus_plus_num(M),Na)) ).

% semiring_norm(6)
tff(fact_261_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),plus_plus_complex(B,C)) = plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_262_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(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_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_263_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_264_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_265_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_266_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] : times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,V),plus_p5714425477246183910nteger(B,C)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,V),B),times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,V),C)) ).

% distrib_left_numeral
tff(fact_267_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(plus_plus_complex(A,B)),numera6690914467698888265omplex(V)) = plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_268_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),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_269_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_270_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_271_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_272_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : times_3573771949741848930nteger(plus_p5714425477246183910nteger(A,B),aa_num_Code_integer(numera6620942414471956472nteger,V)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,V)),times_3573771949741848930nteger(B,aa_num_Code_integer(numera6620942414471956472nteger,V))) ).

% distrib_right_numeral
tff(fact_273_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_274_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : plus_plus_complex(numera6690914467698888265omplex(M),numera6690914467698888265omplex(Na)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),Na)) ).

% numeral_plus_numeral
tff(fact_275_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),Na)) ).

% numeral_plus_numeral
tff(fact_276_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),Na)) ).

% numeral_plus_numeral
tff(fact_277_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Na)) ).

% numeral_plus_numeral
tff(fact_278_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),Na)) ).

% numeral_plus_numeral
tff(fact_279_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),Na)) ).

% numeral_plus_numeral
tff(fact_280_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : plus_plus_complex(numera6690914467698888265omplex(V),plus_plus_complex(numera6690914467698888265omplex(W),Z)) = plus_plus_complex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W)),Z) ).

% add_numeral_left
tff(fact_281_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_282_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_283_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(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_284_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(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_285_add__numeral__left,axiom,
    ! [V: num,W: num,Z: code_integer] : plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,V),plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,W),Z)) = plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(V),W)),Z) ).

% add_numeral_left
tff(fact_286_semiring__norm_I2_J,axiom,
    aa_num_num(plus_plus_num(one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_287_semiring__norm_I16_J,axiom,
    ! [M: num,Na: num] : aa_num_num(times_times_num(aa_num_num(bit1,M)),aa_num_num(bit1,Na)) = aa_num_num(bit1,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(M),Na)),aa_num_num(bit0,aa_num_num(times_times_num(M),Na)))) ).

% semiring_norm(16)
tff(fact_288_semiring__norm_I9_J,axiom,
    ! [M: num,Na: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),aa_num_num(bit0,Na)) = aa_num_num(bit1,aa_num_num(plus_plus_num(M),Na)) ).

% semiring_norm(9)
tff(fact_289_semiring__norm_I7_J,axiom,
    ! [M: num,Na: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),aa_num_num(bit1,Na)) = aa_num_num(bit1,aa_num_num(plus_plus_num(M),Na)) ).

% semiring_norm(7)
tff(fact_290_semiring__norm_I10_J,axiom,
    ! [M: num,Na: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),aa_num_num(bit1,Na)) = aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(M),Na)),one)) ).

% semiring_norm(10)
tff(fact_291_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(plus_plus_num(M),one)) ).

% semiring_norm(8)
tff(fact_292_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_293_int__int__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( M = Na ) ) ).

% int_int_eq
tff(fact_294_int__distrib_I1_J,axiom,
    ! [Z1: int,Z2: 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),Z2)),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,Z2),W)) ).

% int_distrib(1)
tff(fact_295_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z2: 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),Z2)) = 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),Z2)) ).

% int_distrib(2)
tff(fact_296_add__One__commute,axiom,
    ! [Na: num] : aa_num_num(plus_plus_num(one),Na) = aa_num_num(plus_plus_num(Na),one) ).

% add_One_commute
tff(fact_297_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% is_num_normalize(1)
tff(fact_298_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% is_num_normalize(1)
tff(fact_299_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% is_num_normalize(1)
tff(fact_300_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% nat_neq_iff
tff(fact_301_less__not__refl,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_not_refl
tff(fact_302_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_303_less__not__refl3,axiom,
    ! [Sa: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Sa),T)
     => ( Sa != T ) ) ).

% less_not_refl3
tff(fact_304_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_irrefl_nat
tff(fact_305_nat__less__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% nat_less_induct
tff(fact_306_infinite__descent,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & ~ aa_nat_o(P,M2) ) )
     => aa_nat_o(P,Na) ) ).

% infinite_descent
tff(fact_307_linorder__neqE__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_308_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Xa)),Y) = aa_real_real(times_times_real(Y),aa_nat_real(semiri5074537144036343181t_real,Xa)) ).

% mult_of_nat_commute
tff(fact_309_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,Xa)) ).

% mult_of_nat_commute
tff(fact_310_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xa)) ).

% mult_of_nat_commute
tff(fact_311_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,Xa)) ).

% mult_of_nat_commute
tff(fact_312_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: code_integer] : times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xa),Y) = times_3573771949741848930nteger(Y,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)) ).

% mult_of_nat_commute
tff(fact_313_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Xa)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,Xa)) ).

% mult_of_nat_commute
tff(fact_314_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_315_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_316_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_317_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_318_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_319_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_less_add1
tff(fact_320_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_less_add2
tff(fact_321_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% less_add_eq_less
tff(fact_322_add__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% add_mult_distrib
tff(fact_323_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% add_mult_distrib2
tff(fact_324_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_325_zadd__int__left,axiom,
    ! [M: nat,Na: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))),Z) ).

% zadd_int_left
tff(fact_326_numeral__Bit0,axiom,
    ! [Na: num] : numera6690914467698888265omplex(aa_num_num(bit0,Na)) = plus_plus_complex(numera6690914467698888265omplex(Na),numera6690914467698888265omplex(Na)) ).

% numeral_Bit0
tff(fact_327_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),aa_num_real(numeral_numeral_real,Na)) ).

% numeral_Bit0
tff(fact_328_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_num_rat(numeral_numeral_rat,Na)) ).

% numeral_Bit0
tff(fact_329_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Na)) ).

% numeral_Bit0
tff(fact_330_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Na)) ).

% numeral_Bit0
tff(fact_331_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na)) = plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% numeral_Bit0
tff(fact_332_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_333_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,one)),A) = A ).

% mult_numeral_1
tff(fact_334_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_335_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A) = A ).

% mult_numeral_1
tff(fact_336_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_337_mult__numeral__1,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),A) = A ).

% mult_numeral_1
tff(fact_338_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_339_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,one)) = A ).

% mult_numeral_1_right
tff(fact_340_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_341_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,one)) = A ).

% mult_numeral_1_right
tff(fact_342_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_343_mult__numeral__1__right,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,one)) = A ).

% mult_numeral_1_right
tff(fact_344_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X2: num] : Y != aa_num_num(bit0,X2)
       => ~ ! [X3: num] : Y != aa_num_num(bit1,X3) ) ) ).

% num.exhaust
tff(fact_345_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ).

% less_imp_of_nat_less
tff(fact_346_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% less_imp_of_nat_less
tff(fact_347_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_348_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_349_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ) ).

% less_imp_of_nat_less
tff(fact_350_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_351_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_352_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_353_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_354_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_355_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_356_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_357_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Na)) = $let(
        m: complex,
        m:= numera6690914467698888265omplex(Na),
        plus_plus_complex(m,m) ) ).

% numeral_code(2)
tff(fact_358_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Na)) = $let(
        m: real,
        m:= aa_num_real(numeral_numeral_real,Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m) ) ).

% numeral_code(2)
tff(fact_359_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Na)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,Na),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m),m) ) ).

% numeral_code(2)
tff(fact_360_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,Na),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_361_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Na),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m) ) ).

% numeral_code(2)
tff(fact_362_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,Na),
        plus_p5714425477246183910nteger(m,m) ) ).

% numeral_code(2)
tff(fact_363_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = plus_plus_complex(Z,Z) ).

% mult_2
tff(fact_364_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(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_365_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2
tff(fact_366_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_367_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_368_mult__2,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),Z) = plus_p5714425477246183910nteger(Z,Z) ).

% mult_2
tff(fact_369_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = plus_plus_complex(Z,Z) ).

% mult_2_right
tff(fact_370_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(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_371_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2_right
tff(fact_372_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_373_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_374_mult__2__right,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(Z,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(Z,Z) ).

% mult_2_right
tff(fact_375_left__add__twice,axiom,
    ! [A: complex,B: complex] : plus_plus_complex(A,plus_plus_complex(A,B)) = plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A),B) ).

% left_add_twice
tff(fact_376_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_377_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_378_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_379_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_380_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),A),B) ).

% left_add_twice
tff(fact_381_log2__of__power__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_382_nat__add__offset__less,axiom,
    ! [Y: nat,Na: nat,Xa: nat,M: nat,Sz: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) )
         => 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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),Y)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_383_singleton__conv,axiom,
    ! [A: vEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_af(A)) = insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv
tff(fact_384_singleton__conv,axiom,
    ! [A: complex] : collect_complex(aTP_Lamm_ag(A)) = insert_complex(A,bot_bot_set_complex) ).

% singleton_conv
tff(fact_385_singleton__conv,axiom,
    ! [A: product_prod_int_int] : collec213857154873943460nt_int(aTP_Lamm_ah(A)) = insert5033312907999012233nt_int(A,bot_bo1796632182523588997nt_int) ).

% singleton_conv
tff(fact_386_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(aTP_Lamm_ai,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_387_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_388_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_ak,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_389_singleton__conv2,axiom,
    ! [A: vEBT_VEBT] : collect_VEBT_VEBT(fequal_VEBT_VEBT(A)) = insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv2
tff(fact_390_singleton__conv2,axiom,
    ! [A: complex] : collect_complex(fequal_complex(A)) = insert_complex(A,bot_bot_set_complex) ).

% singleton_conv2
tff(fact_391_singleton__conv2,axiom,
    ! [A: product_prod_int_int] : collec213857154873943460nt_int(fequal6010348034681699726nt_int(A)) = insert5033312907999012233nt_int(A,bot_bo1796632182523588997nt_int) ).

% singleton_conv2
tff(fact_392_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(fequal_real,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_393_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_394_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(fequal_int,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv2
tff(fact_395_Un__insert__left,axiom,
    ! [A: int,B2: set_int,C2: set_int] : sup_sup_set_int(insert_int(A,B2),C2) = insert_int(A,sup_sup_set_int(B2,C2)) ).

% Un_insert_left
tff(fact_396_Un__insert__left,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,C2: set_VEBT_VEBT] : sup_su6272177626956685416T_VEBT(insert_VEBT_VEBT(A,B2),C2) = insert_VEBT_VEBT(A,sup_su6272177626956685416T_VEBT(B2,C2)) ).

% Un_insert_left
tff(fact_397_Un__insert__left,axiom,
    ! [A: real,B2: set_real,C2: set_real] : sup_sup_set_real(insert_real(A,B2),C2) = insert_real(A,sup_sup_set_real(B2,C2)) ).

% Un_insert_left
tff(fact_398_Un__insert__left,axiom,
    ! [A: nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A,B2)),C2) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ).

% Un_insert_left
tff(fact_399_Un__insert__right,axiom,
    ! [A2: set_int,A: int,B2: set_int] : sup_sup_set_int(A2,insert_int(A,B2)) = insert_int(A,sup_sup_set_int(A2,B2)) ).

% Un_insert_right
tff(fact_400_Un__insert__right,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] : sup_su6272177626956685416T_VEBT(A2,insert_VEBT_VEBT(A,B2)) = insert_VEBT_VEBT(A,sup_su6272177626956685416T_VEBT(A2,B2)) ).

% Un_insert_right
tff(fact_401_Un__insert__right,axiom,
    ! [A2: set_real,A: real,B2: set_real] : sup_sup_set_real(A2,insert_real(A,B2)) = insert_real(A,sup_sup_set_real(A2,B2)) ).

% Un_insert_right
tff(fact_402_Un__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),insert_nat(A,B2)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% Un_insert_right
tff(fact_403_Un__empty,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( sup_sup_set_real(A2,B2) = bot_bot_set_real )
    <=> ( ( A2 = bot_bot_set_real )
        & ( B2 = bot_bot_set_real ) ) ) ).

% Un_empty
tff(fact_404_Un__empty,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_405_Un__empty,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( sup_sup_set_int(A2,B2) = bot_bot_set_int )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_406_sup__bot__left,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,bot_bo4199563552545308370d_enat),Xa) = Xa ).

% sup_bot_left
tff(fact_407_sup__bot__left,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),Xa) = Xa ).

% sup_bot_left
tff(fact_408_sup__bot__left,axiom,
    ! [Xa: filter_nat] : sup_sup_filter_nat(bot_bot_filter_nat,Xa) = Xa ).

% sup_bot_left
tff(fact_409_sup__bot__left,axiom,
    ! [Xa: set_real] : sup_sup_set_real(bot_bot_set_real,Xa) = Xa ).

% sup_bot_left
tff(fact_410_sup__bot__left,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),Xa) = Xa ).

% sup_bot_left
tff(fact_411_sup__bot__left,axiom,
    ! [Xa: set_int] : sup_sup_set_int(bot_bot_set_int,Xa) = Xa ).

% sup_bot_left
tff(fact_412_sup__bot__right,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),bot_bo4199563552545308370d_enat) = Xa ).

% sup_bot_right
tff(fact_413_sup__bot__right,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),bot_bot_assn) = Xa ).

% sup_bot_right
tff(fact_414_sup__bot__right,axiom,
    ! [Xa: filter_nat] : sup_sup_filter_nat(Xa,bot_bot_filter_nat) = Xa ).

% sup_bot_right
tff(fact_415_sup__bot__right,axiom,
    ! [Xa: set_real] : sup_sup_set_real(Xa,bot_bot_set_real) = Xa ).

% sup_bot_right
tff(fact_416_sup__bot__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),bot_bot_set_nat) = Xa ).

% sup_bot_right
tff(fact_417_sup__bot__right,axiom,
    ! [Xa: set_int] : sup_sup_set_int(Xa,bot_bot_set_int) = Xa ).

% sup_bot_right
tff(fact_418_bot__eq__sup__iff,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ( bot_bo4199563552545308370d_enat = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) )
    <=> ( ( Xa = bot_bo4199563552545308370d_enat )
        & ( Y = bot_bo4199563552545308370d_enat ) ) ) ).

% bot_eq_sup_iff
tff(fact_419_bot__eq__sup__iff,axiom,
    ! [Xa: assn,Y: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) )
    <=> ( ( Xa = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
tff(fact_420_bot__eq__sup__iff,axiom,
    ! [Xa: filter_nat,Y: filter_nat] :
      ( ( bot_bot_filter_nat = sup_sup_filter_nat(Xa,Y) )
    <=> ( ( Xa = bot_bot_filter_nat )
        & ( Y = bot_bot_filter_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_421_bot__eq__sup__iff,axiom,
    ! [Xa: set_real,Y: set_real] :
      ( ( bot_bot_set_real = sup_sup_set_real(Xa,Y) )
    <=> ( ( Xa = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% bot_eq_sup_iff
tff(fact_422_bot__eq__sup__iff,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) )
    <=> ( ( Xa = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_423_bot__eq__sup__iff,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( bot_bot_set_int = sup_sup_set_int(Xa,Y) )
    <=> ( ( Xa = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
tff(fact_424_sup__eq__bot__iff,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) = bot_bo4199563552545308370d_enat )
    <=> ( ( Xa = bot_bo4199563552545308370d_enat )
        & ( Y = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_eq_bot_iff
tff(fact_425_sup__eq__bot__iff,axiom,
    ! [Xa: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) = bot_bot_assn )
    <=> ( ( Xa = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
tff(fact_426_sup__eq__bot__iff,axiom,
    ! [Xa: filter_nat,Y: filter_nat] :
      ( ( sup_sup_filter_nat(Xa,Y) = bot_bot_filter_nat )
    <=> ( ( Xa = bot_bot_filter_nat )
        & ( Y = bot_bot_filter_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_427_sup__eq__bot__iff,axiom,
    ! [Xa: set_real,Y: set_real] :
      ( ( sup_sup_set_real(Xa,Y) = bot_bot_set_real )
    <=> ( ( Xa = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% sup_eq_bot_iff
tff(fact_428_sup__eq__bot__iff,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) = bot_bot_set_nat )
    <=> ( ( Xa = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_429_sup__eq__bot__iff,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( sup_sup_set_int(Xa,Y) = bot_bot_set_int )
    <=> ( ( Xa = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
tff(fact_430_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = bot_bo4199563552545308370d_enat )
    <=> ( ( A = bot_bo4199563552545308370d_enat )
        & ( B = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_431_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = bot_bot_assn )
    <=> ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_432_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( sup_sup_filter_nat(A,B) = bot_bot_filter_nat )
    <=> ( ( A = bot_bot_filter_nat )
        & ( B = bot_bot_filter_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_433_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( sup_sup_set_real(A,B) = bot_bot_set_real )
    <=> ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_434_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_435_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( sup_sup_set_int(A,B) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_436_sup__bot_Oleft__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,bot_bo4199563552545308370d_enat),A) = A ).

% sup_bot.left_neutral
tff(fact_437_sup__bot_Oleft__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),A) = A ).

% sup_bot.left_neutral
tff(fact_438_sup__bot_Oleft__neutral,axiom,
    ! [A: filter_nat] : sup_sup_filter_nat(bot_bot_filter_nat,A) = A ).

% sup_bot.left_neutral
tff(fact_439_sup__bot_Oleft__neutral,axiom,
    ! [A: set_real] : sup_sup_set_real(bot_bot_set_real,A) = A ).

% sup_bot.left_neutral
tff(fact_440_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_441_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] : sup_sup_set_int(bot_bot_set_int,A) = A ).

% sup_bot.left_neutral
tff(fact_442_empty__Collect__eq,axiom,
    ! [P: fun_complex_o] :
      ( ( bot_bot_set_complex = collect_complex(P) )
    <=> ! [X4: complex] : ~ aa_complex_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_443_empty__Collect__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( ( bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(P) )
    <=> ! [X4: product_prod_int_int] : ~ aa_Pro4592068669816069595_int_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_444_empty__Collect__eq,axiom,
    ! [P: fun_real_o] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X4: real] : ~ aa_real_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_445_empty__Collect__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_446_empty__Collect__eq,axiom,
    ! [P: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_447_Collect__empty__eq,axiom,
    ! [P: fun_complex_o] :
      ( ( collect_complex(P) = bot_bot_set_complex )
    <=> ! [X4: complex] : ~ aa_complex_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_448_Collect__empty__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( ( collec213857154873943460nt_int(P) = bot_bo1796632182523588997nt_int )
    <=> ! [X4: product_prod_int_int] : ~ aa_Pro4592068669816069595_int_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_449_Collect__empty__eq,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X4: real] : ~ aa_real_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_450_Collect__empty__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_451_Collect__empty__eq,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_452_all__not__in__conv,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),A2)
    <=> ( A2 = bot_bo8194388402131092736T_VEBT ) ) ).

% all_not_in_conv
tff(fact_453_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] : ~ aa_set_real_o(member_real(X4),A2)
    <=> ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_454_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ~ aa_set_nat_o(member_nat(X4),A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_455_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ~ aa_set_int_o(member_int(X4),A2)
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_456_empty__iff,axiom,
    ! [C: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),bot_bo8194388402131092736T_VEBT) ).

% empty_iff
tff(fact_457_empty__iff,axiom,
    ! [C: real] : ~ aa_set_real_o(member_real(C),bot_bot_set_real) ).

% empty_iff
tff(fact_458_empty__iff,axiom,
    ! [C: nat] : ~ aa_set_nat_o(member_nat(C),bot_bot_set_nat) ).

% empty_iff
tff(fact_459_empty__iff,axiom,
    ! [C: int] : ~ aa_set_int_o(member_int(C),bot_bot_set_int) ).

% empty_iff
tff(fact_460_insert__absorb2,axiom,
    ! [Xa: nat,A2: set_nat] : insert_nat(Xa,insert_nat(Xa,A2)) = insert_nat(Xa,A2) ).

% insert_absorb2
tff(fact_461_insert__absorb2,axiom,
    ! [Xa: int,A2: set_int] : insert_int(Xa,insert_int(Xa,A2)) = insert_int(Xa,A2) ).

% insert_absorb2
tff(fact_462_insert__absorb2,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT] : insert_VEBT_VEBT(Xa,insert_VEBT_VEBT(Xa,A2)) = insert_VEBT_VEBT(Xa,A2) ).

% insert_absorb2
tff(fact_463_insert__absorb2,axiom,
    ! [Xa: real,A2: set_real] : insert_real(Xa,insert_real(Xa,A2)) = insert_real(Xa,A2) ).

% insert_absorb2
tff(fact_464_insert__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A2))
    <=> ( ( A = B )
        | aa_set_nat_o(member_nat(A),A2) ) ) ).

% insert_iff
tff(fact_465_insert__iff,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B,A2))
    <=> ( ( A = B )
        | aa_set_real_o(member_real(A),A2) ) ) ).

% insert_iff
tff(fact_466_insert__iff,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A2))
    <=> ( ( A = B )
        | aa_set_int_o(member_int(A),A2) ) ) ).

% insert_iff
tff(fact_467_insert__iff,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),insert_VEBT_VEBT(B,A2))
    <=> ( ( A = B )
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2) ) ) ).

% insert_iff
tff(fact_468_insertCI,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( ~ aa_set_nat_o(member_nat(A),B2)
       => ( A = B ) )
     => aa_set_nat_o(member_nat(A),insert_nat(B,B2)) ) ).

% insertCI
tff(fact_469_insertCI,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( ~ aa_set_real_o(member_real(A),B2)
       => ( A = B ) )
     => aa_set_real_o(member_real(A),insert_real(B,B2)) ) ).

% insertCI
tff(fact_470_insertCI,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( ~ aa_set_int_o(member_int(A),B2)
       => ( A = B ) )
     => aa_set_int_o(member_int(A),insert_int(B,B2)) ) ).

% insertCI
tff(fact_471_insertCI,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B2)
       => ( A = B ) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),insert_VEBT_VEBT(B,B2)) ) ).

% insertCI
tff(fact_472_sup_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% sup.right_idem
tff(fact_473_sup_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ).

% sup.right_idem
tff(fact_474_sup_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ).

% sup.right_idem
tff(fact_475_sup_Oright__idem,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) ).

% sup.right_idem
tff(fact_476_sup_Oright__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ).

% sup.right_idem
tff(fact_477_sup__left__idem,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) ).

% sup_left_idem
tff(fact_478_sup__left__idem,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y) ).

% sup_left_idem
tff(fact_479_sup__left__idem,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y) ).

% sup_left_idem
tff(fact_480_sup__left__idem,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) ).

% sup_left_idem
tff(fact_481_sup__left__idem,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) ).

% sup_left_idem
tff(fact_482_sup_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% sup.left_idem
tff(fact_483_sup_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ).

% sup.left_idem
tff(fact_484_sup_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ).

% sup.left_idem
tff(fact_485_sup_Oleft__idem,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) ).

% sup.left_idem
tff(fact_486_sup_Oleft__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ).

% sup.left_idem
tff(fact_487_sup__idem,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Xa) = Xa ).

% sup_idem
tff(fact_488_sup__idem,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Xa) = Xa ).

% sup_idem
tff(fact_489_sup__idem,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Xa) = Xa ).

% sup_idem
tff(fact_490_sup__idem,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Xa) = Xa ).

% sup_idem
tff(fact_491_sup__idem,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Xa) = Xa ).

% sup_idem
tff(fact_492_sup_Oidem,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),A) = A ).

% sup.idem
tff(fact_493_sup_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),A) = A ).

% sup.idem
tff(fact_494_sup_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,A),A) = A ).

% sup.idem
tff(fact_495_sup_Oidem,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),A) = A ).

% sup.idem
tff(fact_496_sup_Oidem,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),A) = A ).

% sup.idem
tff(fact_497_Un__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2))
    <=> ( aa_set_real_o(member_real(C),A2)
        | aa_set_real_o(member_real(C),B2) ) ) ).

% Un_iff
tff(fact_498_Un__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2))
    <=> ( aa_set_int_o(member_int(C),A2)
        | aa_set_int_o(member_int(C),B2) ) ) ).

% Un_iff
tff(fact_499_Un__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),sup_su6272177626956685416T_VEBT(A2,B2))
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ) ).

% Un_iff
tff(fact_500_Un__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
    <=> ( aa_set_nat_o(member_nat(C),A2)
        | aa_set_nat_o(member_nat(C),B2) ) ) ).

% Un_iff
tff(fact_501_UnCI,axiom,
    ! [C: real,B2: set_real,A2: set_real] :
      ( ( ~ aa_set_real_o(member_real(C),B2)
       => aa_set_real_o(member_real(C),A2) )
     => aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2)) ) ).

% UnCI
tff(fact_502_UnCI,axiom,
    ! [C: int,B2: set_int,A2: set_int] :
      ( ( ~ aa_set_int_o(member_int(C),B2)
       => aa_set_int_o(member_int(C),A2) )
     => aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2)) ) ).

% UnCI
tff(fact_503_UnCI,axiom,
    ! [C: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),sup_su6272177626956685416T_VEBT(A2,B2)) ) ).

% UnCI
tff(fact_504_UnCI,axiom,
    ! [C: nat,B2: set_nat,A2: set_nat] :
      ( ( ~ aa_set_nat_o(member_nat(C),B2)
       => aa_set_nat_o(member_nat(C),A2) )
     => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% UnCI
tff(fact_505_singletonI,axiom,
    ! [A: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)) ).

% singletonI
tff(fact_506_singletonI,axiom,
    ! [A: real] : aa_set_real_o(member_real(A),insert_real(A,bot_bot_set_real)) ).

% singletonI
tff(fact_507_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(member_nat(A),insert_nat(A,bot_bot_set_nat)) ).

% singletonI
tff(fact_508_singletonI,axiom,
    ! [A: int] : aa_set_int_o(member_int(A),insert_int(A,bot_bot_set_int)) ).

% singletonI
tff(fact_509_sup__bot_Oright__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),bot_bo4199563552545308370d_enat) = A ).

% sup_bot.right_neutral
tff(fact_510_sup__bot_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),bot_bot_assn) = A ).

% sup_bot.right_neutral
tff(fact_511_sup__bot_Oright__neutral,axiom,
    ! [A: filter_nat] : sup_sup_filter_nat(A,bot_bot_filter_nat) = A ).

% sup_bot.right_neutral
tff(fact_512_sup__bot_Oright__neutral,axiom,
    ! [A: set_real] : sup_sup_set_real(A,bot_bot_set_real) = A ).

% sup_bot.right_neutral
tff(fact_513_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),bot_bot_set_nat) = A ).

% sup_bot.right_neutral
tff(fact_514_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] : sup_sup_set_int(A,bot_bot_set_int) = A ).

% sup_bot.right_neutral
tff(fact_515_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( bot_bo4199563552545308370d_enat = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) )
    <=> ( ( A = bot_bo4199563552545308370d_enat )
        & ( B = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_516_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
    <=> ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_517_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( bot_bot_filter_nat = sup_sup_filter_nat(A,B) )
    <=> ( ( A = bot_bot_filter_nat )
        & ( B = bot_bot_filter_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_518_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( bot_bot_set_real = sup_sup_set_real(A,B) )
    <=> ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_519_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_520_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int = sup_sup_set_int(A,B) )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_521_bot__set__def,axiom,
    bot_bot_set_complex = collect_complex(bot_bo8750099748576881071plex_o) ).

% bot_set_def
tff(fact_522_bot__set__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(bot_bo3516217578726220278_int_o) ).

% bot_set_def
tff(fact_523_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bot_fun_real_o) ).

% bot_set_def
tff(fact_524_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_525_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_526_sup__set__def,axiom,
    ! [A2: set_real,B2: set_real] : sup_sup_set_real(A2,B2) = collect_real(sup_sup_fun_real_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).

% sup_set_def
tff(fact_527_sup__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : sup_su6272177626956685416T_VEBT(A2,B2) = collect_VEBT_VEBT(sup_su3075289160232154067VEBT_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% sup_set_def
tff(fact_528_sup__set__def,axiom,
    ! [A2: set_int,B2: set_int] : sup_sup_set_int(A2,B2) = collect_int(sup_sup_fun_int_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).

% sup_set_def
tff(fact_529_sup__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] : sup_sup_set_complex(A2,B2) = collect_complex(sup_su37214000559163671plex_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2))) ).

% sup_set_def
tff(fact_530_sup__set__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : sup_su6024340866399070445nt_int(A2,B2) = collec213857154873943460nt_int(sup_su4047618219528032398_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% sup_set_def
tff(fact_531_sup__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = collect_nat(sup_sup_fun_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% sup_set_def
tff(fact_532_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ aa_set_real_o(ord_less_set_real(A2),bot_bot_set_real) ).

% not_psubset_empty
tff(fact_533_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A2),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_534_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ aa_set_int_o(ord_less_set_int(A2),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_535_ex__in__conv,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ? [X4: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),A2)
    <=> ( A2 != bot_bo8194388402131092736T_VEBT ) ) ).

% ex_in_conv
tff(fact_536_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ? [X4: real] : aa_set_real_o(member_real(X4),A2)
    <=> ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_537_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X4: nat] : aa_set_nat_o(member_nat(X4),A2)
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_538_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X4: int] : aa_set_int_o(member_int(X4),A2)
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_539_equals0I,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ! [Y2: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y2),A2)
     => ( A2 = bot_bo8194388402131092736T_VEBT ) ) ).

% equals0I
tff(fact_540_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y2: real] : ~ aa_set_real_o(member_real(Y2),A2)
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_541_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y2: nat] : ~ aa_set_nat_o(member_nat(Y2),A2)
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_542_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y2: int] : ~ aa_set_int_o(member_int(Y2),A2)
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_543_equals0D,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( A2 = bot_bo8194388402131092736T_VEBT )
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2) ) ).

% equals0D
tff(fact_544_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ aa_set_real_o(member_real(A),A2) ) ).

% equals0D
tff(fact_545_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A),A2) ) ).

% equals0D
tff(fact_546_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A),A2) ) ).

% equals0D
tff(fact_547_emptyE,axiom,
    ! [A: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ).

% emptyE
tff(fact_548_emptyE,axiom,
    ! [A: real] : ~ aa_set_real_o(member_real(A),bot_bot_set_real) ).

% emptyE
tff(fact_549_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat(A),bot_bot_set_nat) ).

% emptyE
tff(fact_550_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).

% emptyE
tff(fact_551_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ? [B3: set_nat] :
          ( ( A2 = insert_nat(A,B3) )
          & ~ aa_set_nat_o(member_nat(A),B3) ) ) ).

% mk_disjoint_insert
tff(fact_552_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ? [B3: set_real] :
          ( ( A2 = insert_real(A,B3) )
          & ~ aa_set_real_o(member_real(A),B3) ) ) ).

% mk_disjoint_insert
tff(fact_553_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ? [B3: set_int] :
          ( ( A2 = insert_int(A,B3) )
          & ~ aa_set_int_o(member_int(A),B3) ) ) ).

% mk_disjoint_insert
tff(fact_554_mk__disjoint__insert,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
     => ? [B3: set_VEBT_VEBT] :
          ( ( A2 = insert_VEBT_VEBT(A,B3) )
          & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B3) ) ) ).

% mk_disjoint_insert
tff(fact_555_insert__commute,axiom,
    ! [Xa: nat,Y: nat,A2: set_nat] : insert_nat(Xa,insert_nat(Y,A2)) = insert_nat(Y,insert_nat(Xa,A2)) ).

% insert_commute
tff(fact_556_insert__commute,axiom,
    ! [Xa: int,Y: int,A2: set_int] : insert_int(Xa,insert_int(Y,A2)) = insert_int(Y,insert_int(Xa,A2)) ).

% insert_commute
tff(fact_557_insert__commute,axiom,
    ! [Xa: vEBT_VEBT,Y: vEBT_VEBT,A2: set_VEBT_VEBT] : insert_VEBT_VEBT(Xa,insert_VEBT_VEBT(Y,A2)) = insert_VEBT_VEBT(Y,insert_VEBT_VEBT(Xa,A2)) ).

% insert_commute
tff(fact_558_insert__commute,axiom,
    ! [Xa: real,Y: real,A2: set_real] : insert_real(Xa,insert_real(Y,A2)) = insert_real(Y,insert_real(Xa,A2)) ).

% insert_commute
tff(fact_559_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A2)
     => ( ~ aa_set_nat_o(member_nat(B),B2)
       => ( ( insert_nat(A,A2) = insert_nat(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C3: set_nat] :
                ( ( A2 = insert_nat(B,C3) )
                & ~ aa_set_nat_o(member_nat(B),C3)
                & ( B2 = insert_nat(A,C3) )
                & ~ aa_set_nat_o(member_nat(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_560_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B: real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(A),A2)
     => ( ~ aa_set_real_o(member_real(B),B2)
       => ( ( insert_real(A,A2) = insert_real(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C3: set_real] :
                ( ( A2 = insert_real(B,C3) )
                & ~ aa_set_real_o(member_real(B),C3)
                & ( B2 = insert_real(A,C3) )
                & ~ aa_set_real_o(member_real(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_561_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B: int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(A),A2)
     => ( ~ aa_set_int_o(member_int(B),B2)
       => ( ( insert_int(A,A2) = insert_int(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C3: set_int] :
                ( ( A2 = insert_int(B,C3) )
                & ~ aa_set_int_o(member_int(B),C3)
                & ( B2 = insert_int(A,C3) )
                & ~ aa_set_int_o(member_int(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_562_insert__eq__iff,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),B2)
       => ( ( insert_VEBT_VEBT(A,A2) = insert_VEBT_VEBT(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C3: set_VEBT_VEBT] :
                ( ( A2 = insert_VEBT_VEBT(B,C3) )
                & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),C3)
                & ( B2 = insert_VEBT_VEBT(A,C3) )
                & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_563_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( insert_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_564_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( insert_real(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_565_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_566_insert__absorb,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
     => ( insert_VEBT_VEBT(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_567_insert__ident,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( ~ aa_set_nat_o(member_nat(Xa),B2)
       => ( ( insert_nat(Xa,A2) = insert_nat(Xa,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_568_insert__ident,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( ~ aa_set_real_o(member_real(Xa),B2)
       => ( ( insert_real(Xa,A2) = insert_real(Xa,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_569_insert__ident,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( ~ aa_set_int_o(member_int(Xa),B2)
       => ( ( insert_int(Xa,A2) = insert_int(Xa,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_570_insert__ident,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B2)
       => ( ( insert_VEBT_VEBT(Xa,A2) = insert_VEBT_VEBT(Xa,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_571_Set_Oset__insert,axiom,
    ! [Xa: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),A2)
     => ~ ! [B3: set_nat] :
            ( ( A2 = insert_nat(Xa,B3) )
           => aa_set_nat_o(member_nat(Xa),B3) ) ) ).

% Set.set_insert
tff(fact_572_Set_Oset__insert,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_set_real_o(member_real(Xa),A2)
     => ~ ! [B3: set_real] :
            ( ( A2 = insert_real(Xa,B3) )
           => aa_set_real_o(member_real(Xa),B3) ) ) ).

% Set.set_insert
tff(fact_573_Set_Oset__insert,axiom,
    ! [Xa: int,A2: set_int] :
      ( aa_set_int_o(member_int(Xa),A2)
     => ~ ! [B3: set_int] :
            ( ( A2 = insert_int(Xa,B3) )
           => aa_set_int_o(member_int(Xa),B3) ) ) ).

% Set.set_insert
tff(fact_574_Set_Oset__insert,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2)
     => ~ ! [B3: set_VEBT_VEBT] :
            ( ( A2 = insert_VEBT_VEBT(Xa,B3) )
           => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B3) ) ) ).

% Set.set_insert
tff(fact_575_insertI2,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( aa_set_nat_o(member_nat(A),B2)
     => aa_set_nat_o(member_nat(A),insert_nat(B,B2)) ) ).

% insertI2
tff(fact_576_insertI2,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( aa_set_real_o(member_real(A),B2)
     => aa_set_real_o(member_real(A),insert_real(B,B2)) ) ).

% insertI2
tff(fact_577_insertI2,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( aa_set_int_o(member_int(A),B2)
     => aa_set_int_o(member_int(A),insert_int(B,B2)) ) ).

% insertI2
tff(fact_578_insertI2,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B2)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),insert_VEBT_VEBT(B,B2)) ) ).

% insertI2
tff(fact_579_insertI1,axiom,
    ! [A: nat,B2: set_nat] : aa_set_nat_o(member_nat(A),insert_nat(A,B2)) ).

% insertI1
tff(fact_580_insertI1,axiom,
    ! [A: real,B2: set_real] : aa_set_real_o(member_real(A),insert_real(A,B2)) ).

% insertI1
tff(fact_581_insertI1,axiom,
    ! [A: int,B2: set_int] : aa_set_int_o(member_int(A),insert_int(A,B2)) ).

% insertI1
tff(fact_582_insertI1,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),insert_VEBT_VEBT(A,B2)) ).

% insertI1
tff(fact_583_insertE,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A2))
     => ( ( A != B )
       => aa_set_nat_o(member_nat(A),A2) ) ) ).

% insertE
tff(fact_584_insertE,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B,A2))
     => ( ( A != B )
       => aa_set_real_o(member_real(A),A2) ) ) ).

% insertE
tff(fact_585_insertE,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A2))
     => ( ( A != B )
       => aa_set_int_o(member_int(A),A2) ) ) ).

% insertE
tff(fact_586_insertE,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),insert_VEBT_VEBT(B,A2))
     => ( ( A != B )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2) ) ) ).

% insertE
tff(fact_587_sup__left__commute,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ).

% sup_left_commute
tff(fact_588_sup__left__commute,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z)) ).

% sup_left_commute
tff(fact_589_sup__left__commute,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Z)) ).

% sup_left_commute
tff(fact_590_sup__left__commute,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Z)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Z)) ).

% sup_left_commute
tff(fact_591_sup__left__commute,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Z)) ).

% sup_left_commute
tff(fact_592_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ).

% sup.left_commute
tff(fact_593_sup_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)) ).

% sup.left_commute
tff(fact_594_sup_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),C)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)) ).

% sup.left_commute
tff(fact_595_sup_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),C)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,B),C)) ).

% sup.left_commute
tff(fact_596_sup_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)) ).

% sup.left_commute
tff(fact_597_sup__commute,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Xa) ).

% sup_commute
tff(fact_598_sup__commute,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Xa) ).

% sup_commute
tff(fact_599_sup__commute,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Xa) ).

% sup_commute
tff(fact_600_sup__commute,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Xa) ).

% sup_commute
tff(fact_601_sup__commute,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Xa) ).

% sup_commute
tff(fact_602_sup_Ocommute,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A) ).

% sup.commute
tff(fact_603_sup_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),A) ).

% sup.commute
tff(fact_604_sup_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = aa_int_int(aa_int_fun_int_int(sup_sup_int,B),A) ).

% sup.commute
tff(fact_605_sup_Ocommute,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,B),A) ).

% sup.commute
tff(fact_606_sup_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),A) ).

% sup.commute
tff(fact_607_sup__assoc,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).

% sup_assoc
tff(fact_608_sup__assoc,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) ).

% sup_assoc
tff(fact_609_sup__assoc,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)),Z) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) ).

% sup_assoc
tff(fact_610_sup__assoc,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)),Z) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Z)) ).

% sup_assoc
tff(fact_611_sup__assoc,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% sup_assoc
tff(fact_612_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ).

% sup.assoc
tff(fact_613_sup_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)) ).

% sup.assoc
tff(fact_614_sup_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)) ).

% sup.assoc
tff(fact_615_sup_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)),C) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,B),C)) ).

% sup.assoc
tff(fact_616_sup_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)) ).

% sup.assoc
tff(fact_617_inf__sup__aci_I5_J,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Xa) ).

% inf_sup_aci(5)
tff(fact_618_inf__sup__aci_I5_J,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Xa) ).

% inf_sup_aci(5)
tff(fact_619_inf__sup__aci_I5_J,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Xa) ).

% inf_sup_aci(5)
tff(fact_620_inf__sup__aci_I5_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Xa) ).

% inf_sup_aci(5)
tff(fact_621_inf__sup__aci_I5_J,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Xa) ).

% inf_sup_aci(5)
tff(fact_622_inf__sup__aci_I6_J,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_623_inf__sup__aci_I6_J,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_624_inf__sup__aci_I6_J,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)),Z) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_625_inf__sup__aci_I6_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)),Z) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_626_inf__sup__aci_I6_J,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_627_inf__sup__aci_I7_J,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_628_inf__sup__aci_I7_J,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_629_inf__sup__aci_I7_J,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_630_inf__sup__aci_I7_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Z)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_631_inf__sup__aci_I7_J,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_632_inf__sup__aci_I8_J,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) ).

% inf_sup_aci(8)
tff(fact_633_inf__sup__aci_I8_J,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y) ).

% inf_sup_aci(8)
tff(fact_634_inf__sup__aci_I8_J,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y) ).

% inf_sup_aci(8)
tff(fact_635_inf__sup__aci_I8_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) ).

% inf_sup_aci(8)
tff(fact_636_inf__sup__aci_I8_J,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) ).

% inf_sup_aci(8)
tff(fact_637_Un__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C2)) ).

% Un_left_commute
tff(fact_638_Un__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ).

% Un_left_absorb
tff(fact_639_Un__commute,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2) ).

% Un_commute
tff(fact_640_Un__absorb,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),A2) = A2 ).

% Un_absorb
tff(fact_641_Un__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ).

% Un_assoc
tff(fact_642_ball__Un,axiom,
    ! [A2: set_nat,B2: set_nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
         => aa_nat_o(P,X4) )
    <=> ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
           => aa_nat_o(P,X4) )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B2)
           => aa_nat_o(P,X4) ) ) ) ).

% ball_Un
tff(fact_643_bex__Un,axiom,
    ! [A2: set_nat,B2: set_nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
          & aa_nat_o(P,X4) )
    <=> ( ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & aa_nat_o(P,X4) )
        | ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B2)
            & aa_nat_o(P,X4) ) ) ) ).

% bex_Un
tff(fact_644_UnI2,axiom,
    ! [C: real,B2: set_real,A2: set_real] :
      ( aa_set_real_o(member_real(C),B2)
     => aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2)) ) ).

% UnI2
tff(fact_645_UnI2,axiom,
    ! [C: int,B2: set_int,A2: set_int] :
      ( aa_set_int_o(member_int(C),B2)
     => aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2)) ) ).

% UnI2
tff(fact_646_UnI2,axiom,
    ! [C: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),sup_su6272177626956685416T_VEBT(A2,B2)) ) ).

% UnI2
tff(fact_647_UnI2,axiom,
    ! [C: nat,B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C),B2)
     => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% UnI2
tff(fact_648_UnI1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),A2)
     => aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2)) ) ).

% UnI1
tff(fact_649_UnI1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2)) ) ).

% UnI1
tff(fact_650_UnI1,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),sup_su6272177626956685416T_VEBT(A2,B2)) ) ).

% UnI1
tff(fact_651_UnI1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),A2)
     => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% UnI1
tff(fact_652_UnE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2))
     => ( ~ aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B2) ) ) ).

% UnE
tff(fact_653_UnE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2))
     => ( ~ aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% UnE
tff(fact_654_UnE,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),sup_su6272177626956685416T_VEBT(A2,B2))
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ) ).

% UnE
tff(fact_655_UnE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
     => ( ~ aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% UnE
tff(fact_656_Set_Oempty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_al) ).

% Set.empty_def
tff(fact_657_Set_Oempty__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(aTP_Lamm_am) ).

% Set.empty_def
tff(fact_658_Set_Oempty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_an) ).

% Set.empty_def
tff(fact_659_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_ao) ).

% Set.empty_def
tff(fact_660_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_ap) ).

% Set.empty_def
tff(fact_661_insert__Collect,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_o] : insert_VEBT_VEBT(A,collect_VEBT_VEBT(P)) = collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_aq(A),P)) ).

% insert_Collect
tff(fact_662_insert__Collect,axiom,
    ! [A: real,P: fun_real_o] : insert_real(A,collect_real(P)) = collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ar(A),P)) ).

% insert_Collect
tff(fact_663_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_o] : insert_nat(A,collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_as(A),P)) ).

% insert_Collect
tff(fact_664_insert__Collect,axiom,
    ! [A: int,P: fun_int_o] : insert_int(A,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_at(A),P)) ).

% insert_Collect
tff(fact_665_insert__Collect,axiom,
    ! [A: complex,P: fun_complex_o] : insert_complex(A,collect_complex(P)) = collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_au(A),P)) ).

% insert_Collect
tff(fact_666_insert__Collect,axiom,
    ! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] : insert5033312907999012233nt_int(A,collec213857154873943460nt_int(P)) = collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_av(A),P)) ).

% insert_Collect
tff(fact_667_insert__compr,axiom,
    ! [A: real,B2: set_real] : insert_real(A,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_aw(A),B2)) ).

% insert_compr
tff(fact_668_insert__compr,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT] : insert_VEBT_VEBT(A,B2) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_ax(A),B2)) ).

% insert_compr
tff(fact_669_insert__compr,axiom,
    ! [A: nat,B2: set_nat] : insert_nat(A,B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ay(A),B2)) ).

% insert_compr
tff(fact_670_insert__compr,axiom,
    ! [A: int,B2: set_int] : insert_int(A,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_az(A),B2)) ).

% insert_compr
tff(fact_671_insert__compr,axiom,
    ! [A: complex,B2: set_complex] : insert_complex(A,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_ba(A),B2)) ).

% insert_compr
tff(fact_672_insert__compr,axiom,
    ! [A: product_prod_int_int,B2: set_Pr958786334691620121nt_int] : insert5033312907999012233nt_int(A,B2) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_bb(A),B2)) ).

% insert_compr
tff(fact_673_Collect__disj__eq,axiom,
    ! [P: fun_int_o,Q: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bc(P),Q)) = sup_sup_set_int(collect_int(P),collect_int(Q)) ).

% Collect_disj_eq
tff(fact_674_Collect__disj__eq,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] : collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_bd(P),Q)) = sup_sup_set_complex(collect_complex(P),collect_complex(Q)) ).

% Collect_disj_eq
tff(fact_675_Collect__disj__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_be(P),Q)) = sup_su6024340866399070445nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q)) ).

% Collect_disj_eq
tff(fact_676_Collect__disj__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bf(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(P)),collect_nat(Q)) ).

% Collect_disj_eq
tff(fact_677_Un__def,axiom,
    ! [A2: set_real,B2: set_real] : sup_sup_set_real(A2,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_bg(A2),B2)) ).

% Un_def
tff(fact_678_Un__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : sup_su6272177626956685416T_VEBT(A2,B2) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_bh(A2),B2)) ).

% Un_def
tff(fact_679_Un__def,axiom,
    ! [A2: set_int,B2: set_int] : sup_sup_set_int(A2,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_bi(A2),B2)) ).

% Un_def
tff(fact_680_Un__def,axiom,
    ! [A2: set_complex,B2: set_complex] : sup_sup_set_complex(A2,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_bj(A2),B2)) ).

% Un_def
tff(fact_681_Un__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : sup_su6024340866399070445nt_int(A2,B2) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_bk(A2),B2)) ).

% Un_def
tff(fact_682_Un__def,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_bl(A2),B2)) ).

% Un_def
tff(fact_683_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(C),B)
     => aa_set_nat_o(ord_less_set_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_684_sup_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),B)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_685_sup_Ostrict__coboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( ord_less_assn(C,B)
     => ord_less_assn(C,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_686_sup_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(sup_sup_real(A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_687_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_688_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_689_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_690_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(C),A)
     => aa_set_nat_o(ord_less_set_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_691_sup_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_692_sup_Ostrict__coboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( ord_less_assn(C,A)
     => ord_less_assn(C,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_693_sup_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(sup_sup_real(A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_694_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_695_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_696_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_697_sup_Ostrict__order__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_698_sup_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
    <=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_699_sup_Ostrict__order__iff,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_assn(B,A)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_700_sup_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(sup_sup_real(A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_701_sup_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_702_sup_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_703_sup_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_704_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
     => ~ ( aa_set_nat_o(ord_less_set_nat(B),A)
         => ~ aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% sup.strict_boundedE
tff(fact_705_sup_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,B),C)),A)
     => ~ ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
         => ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A) ) ) ).

% sup.strict_boundedE
tff(fact_706_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C),A)
     => ~ ( ord_less_assn(B,A)
         => ~ ord_less_assn(C,A) ) ) ).

% sup.strict_boundedE
tff(fact_707_sup_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sup_sup_real(B),C)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_708_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_709_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_710_sup_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_711_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_712_sup_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_713_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_assn(A,B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb4
tff(fact_714_sup_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(sup_sup_real(A),B) = B ) ) ).

% sup.absorb4
tff(fact_715_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_716_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_717_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb4
tff(fact_718_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_719_sup_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_720_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_assn(B,A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb3
tff(fact_721_sup_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(sup_sup_real(A),B) = A ) ) ).

% sup.absorb3
tff(fact_722_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_723_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_724_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb3
tff(fact_725_less__supI2,axiom,
    ! [Xa: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),B)
     => aa_set_nat_o(ord_less_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI2
tff(fact_726_less__supI2,axiom,
    ! [Xa: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Xa),B)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% less_supI2
tff(fact_727_less__supI2,axiom,
    ! [Xa: assn,B: assn,A: assn] :
      ( ord_less_assn(Xa,B)
     => ord_less_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI2
tff(fact_728_less__supI2,axiom,
    ! [Xa: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(sup_sup_real(A),B)) ) ).

% less_supI2
tff(fact_729_less__supI2,axiom,
    ! [Xa: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% less_supI2
tff(fact_730_less__supI2,axiom,
    ! [Xa: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI2
tff(fact_731_less__supI2,axiom,
    ! [Xa: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI2
tff(fact_732_less__supI1,axiom,
    ! [Xa: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),A)
     => aa_set_nat_o(ord_less_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI1
tff(fact_733_less__supI1,axiom,
    ! [Xa: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Xa),A)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% less_supI1
tff(fact_734_less__supI1,axiom,
    ! [Xa: assn,A: assn,B: assn] :
      ( ord_less_assn(Xa,A)
     => ord_less_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI1
tff(fact_735_less__supI1,axiom,
    ! [Xa: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(sup_sup_real(A),B)) ) ).

% less_supI1
tff(fact_736_less__supI1,axiom,
    ! [Xa: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% less_supI1
tff(fact_737_less__supI1,axiom,
    ! [Xa: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI1
tff(fact_738_less__supI1,axiom,
    ! [Xa: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI1
tff(fact_739_singleton__inject,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT] :
      ( ( insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT) = insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_740_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( insert_real(A,bot_bot_set_real) = insert_real(B,bot_bot_set_real) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_741_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( insert_nat(A,bot_bot_set_nat) = insert_nat(B,bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_742_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( insert_int(A,bot_bot_set_int) = insert_int(B,bot_bot_set_int) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_743_insert__not__empty,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] : insert_VEBT_VEBT(A,A2) != bot_bo8194388402131092736T_VEBT ).

% insert_not_empty
tff(fact_744_insert__not__empty,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) != bot_bot_set_real ).

% insert_not_empty
tff(fact_745_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_746_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) != bot_bot_set_int ).

% insert_not_empty
tff(fact_747_doubleton__eq__iff,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,C: vEBT_VEBT,D: vEBT_VEBT] :
      ( ( insert_VEBT_VEBT(A,insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT)) = insert_VEBT_VEBT(C,insert_VEBT_VEBT(D,bot_bo8194388402131092736T_VEBT)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_748_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( insert_real(A,insert_real(B,bot_bot_set_real)) = insert_real(C,insert_real(D,bot_bot_set_real)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_749_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( insert_nat(A,insert_nat(B,bot_bot_set_nat)) = insert_nat(C,insert_nat(D,bot_bot_set_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_750_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( insert_int(A,insert_int(B,bot_bot_set_int)) = insert_int(C,insert_int(D,bot_bot_set_int)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_751_singleton__iff,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_752_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real(B),insert_real(A,bot_bot_set_real))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_753_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),insert_nat(A,bot_bot_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_754_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_755_singletonD,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))
     => ( B = A ) ) ).

% singletonD
tff(fact_756_singletonD,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real(B),insert_real(A,bot_bot_set_real))
     => ( B = A ) ) ).

% singletonD
tff(fact_757_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),insert_nat(A,bot_bot_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_758_singletonD,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
     => ( B = A ) ) ).

% singletonD
tff(fact_759_Un__empty__right,axiom,
    ! [A2: set_real] : sup_sup_set_real(A2,bot_bot_set_real) = A2 ).

% Un_empty_right
tff(fact_760_Un__empty__right,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),bot_bot_set_nat) = A2 ).

% Un_empty_right
tff(fact_761_Un__empty__right,axiom,
    ! [A2: set_int] : sup_sup_set_int(A2,bot_bot_set_int) = A2 ).

% Un_empty_right
tff(fact_762_Un__empty__left,axiom,
    ! [B2: set_real] : sup_sup_set_real(bot_bot_set_real,B2) = B2 ).

% Un_empty_left
tff(fact_763_Un__empty__left,axiom,
    ! [B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B2) = B2 ).

% Un_empty_left
tff(fact_764_Un__empty__left,axiom,
    ! [B2: set_int] : sup_sup_set_int(bot_bot_set_int,B2) = B2 ).

% Un_empty_left
tff(fact_765_Collect__conv__if2,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_bm(A),P)) = $ite(aa_VEBT_VEBT_o(P,A),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if2
tff(fact_766_Collect__conv__if2,axiom,
    ! [A: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_bn(A),P)) = $ite(aa_complex_o(P,A),insert_complex(A,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if2
tff(fact_767_Collect__conv__if2,axiom,
    ! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_bo(A),P)) = $ite(aa_Pro4592068669816069595_int_o(P,A),insert5033312907999012233nt_int(A,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if2
tff(fact_768_Collect__conv__if2,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bp(A),P)) = $ite(aa_real_o(P,A),insert_real(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if2
tff(fact_769_Collect__conv__if2,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bq(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_770_Collect__conv__if2,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_br(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_771_Collect__conv__if,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_bs(A),P)) = $ite(aa_VEBT_VEBT_o(P,A),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if
tff(fact_772_Collect__conv__if,axiom,
    ! [A: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_bt(A),P)) = $ite(aa_complex_o(P,A),insert_complex(A,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if
tff(fact_773_Collect__conv__if,axiom,
    ! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_bu(A),P)) = $ite(aa_Pro4592068669816069595_int_o(P,A),insert5033312907999012233nt_int(A,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if
tff(fact_774_Collect__conv__if,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bv(A),P)) = $ite(aa_real_o(P,A),insert_real(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if
tff(fact_775_Collect__conv__if,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bw(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_776_Collect__conv__if,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bx(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_777_insert__def,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT] : insert_VEBT_VEBT(A,B2) = sup_su6272177626956685416T_VEBT(collect_VEBT_VEBT(aTP_Lamm_af(A)),B2) ).

% insert_def
tff(fact_778_insert__def,axiom,
    ! [A: real,B2: set_real] : insert_real(A,B2) = sup_sup_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_ai,A)),B2) ).

% insert_def
tff(fact_779_insert__def,axiom,
    ! [A: int,B2: set_int] : insert_int(A,B2) = sup_sup_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ak,A)),B2) ).

% insert_def
tff(fact_780_insert__def,axiom,
    ! [A: complex,B2: set_complex] : insert_complex(A,B2) = sup_sup_set_complex(collect_complex(aTP_Lamm_ag(A)),B2) ).

% insert_def
tff(fact_781_insert__def,axiom,
    ! [A: product_prod_int_int,B2: set_Pr958786334691620121nt_int] : insert5033312907999012233nt_int(A,B2) = sup_su6024340866399070445nt_int(collec213857154873943460nt_int(aTP_Lamm_ah(A)),B2) ).

% insert_def
tff(fact_782_insert__def,axiom,
    ! [A: nat,B2: set_nat] : insert_nat(A,B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,A))),B2) ).

% insert_def
tff(fact_783_singleton__Un__iff,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) = sup_su6272177626956685416T_VEBT(A2,B2) )
    <=> ( ( ( A2 = bot_bo8194388402131092736T_VEBT )
          & ( B2 = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) ) )
        | ( ( A2 = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) )
          & ( B2 = bot_bo8194388402131092736T_VEBT ) )
        | ( ( A2 = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) )
          & ( B2 = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) ) ) ) ) ).

% singleton_Un_iff
tff(fact_784_singleton__Un__iff,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( ( insert_real(Xa,bot_bot_set_real) = sup_sup_set_real(A2,B2) )
    <=> ( ( ( A2 = bot_bot_set_real )
          & ( B2 = insert_real(Xa,bot_bot_set_real) ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B2 = insert_real(Xa,bot_bot_set_real) ) ) ) ) ).

% singleton_Un_iff
tff(fact_785_singleton__Un__iff,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( ( insert_nat(Xa,bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B2 = insert_nat(Xa,bot_bot_set_nat) ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B2 = insert_nat(Xa,bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_786_singleton__Un__iff,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( ( insert_int(Xa,bot_bot_set_int) = sup_sup_set_int(A2,B2) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B2 = insert_int(Xa,bot_bot_set_int) ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B2 = insert_int(Xa,bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_787_Un__singleton__iff,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( ( sup_su6272177626956685416T_VEBT(A2,B2) = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) )
    <=> ( ( ( A2 = bot_bo8194388402131092736T_VEBT )
          & ( B2 = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) ) )
        | ( ( A2 = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) )
          & ( B2 = bot_bo8194388402131092736T_VEBT ) )
        | ( ( A2 = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) )
          & ( B2 = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) ) ) ) ) ).

% Un_singleton_iff
tff(fact_788_Un__singleton__iff,axiom,
    ! [A2: set_real,B2: set_real,Xa: real] :
      ( ( sup_sup_set_real(A2,B2) = insert_real(Xa,bot_bot_set_real) )
    <=> ( ( ( A2 = bot_bot_set_real )
          & ( B2 = insert_real(Xa,bot_bot_set_real) ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B2 = insert_real(Xa,bot_bot_set_real) ) ) ) ) ).

% Un_singleton_iff
tff(fact_789_Un__singleton__iff,axiom,
    ! [A2: set_nat,B2: set_nat,Xa: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = insert_nat(Xa,bot_bot_set_nat) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B2 = insert_nat(Xa,bot_bot_set_nat) ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B2 = insert_nat(Xa,bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_790_Un__singleton__iff,axiom,
    ! [A2: set_int,B2: set_int,Xa: int] :
      ( ( sup_sup_set_int(A2,B2) = insert_int(Xa,bot_bot_set_int) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B2 = insert_int(Xa,bot_bot_set_int) ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B2 = insert_int(Xa,bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_791_insert__is__Un,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] : insert_VEBT_VEBT(A,A2) = sup_su6272177626956685416T_VEBT(insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT),A2) ).

% insert_is_Un
tff(fact_792_insert__is__Un,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) = sup_sup_set_real(insert_real(A,bot_bot_set_real),A2) ).

% insert_is_Un
tff(fact_793_insert__is__Un,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A,bot_bot_set_nat)),A2) ).

% insert_is_Un
tff(fact_794_insert__is__Un,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) = sup_sup_set_int(insert_int(A,bot_bot_set_int),A2) ).

% insert_is_Un
tff(fact_795_n__less__equal__power__2,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% n_less_equal_power_2
tff(fact_796_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(2)
tff(fact_797_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_798_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_799_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_800_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_801_add__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,C))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% add_less_cancel_right
tff(fact_802_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_803_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_804_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_805_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_806_add__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(C,A)),plus_p5714425477246183910nteger(C,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% add_less_cancel_left
tff(fact_807_nat__times__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_times_as_int
tff(fact_808_nat__plus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_plus_as_int
tff(fact_809_high__inv,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),Xa)),Na) = Y ) ) ).

% high_inv
tff(fact_810_low__inv,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),Xa)),Na) = Xa ) ) ).

% low_inv
tff(fact_811_mulcomm,axiom,
    ! [I: nat,Va: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),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)))))) = 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))),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)))))),I) ).

% mulcomm
tff(fact_812_pow__sum,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ).

% pow_sum
tff(fact_813_four__x__squared,axiom,
    ! [Xa: 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,Xa),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))),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_814_high__bound__aux,axiom,
    ! [Ma: nat,Na: nat,M: nat] :
      ( 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))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ).

% high_bound_aux
tff(fact_815_bit__split__inv,axiom,
    ! [Xa: 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,Xa),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),D)),D) = Xa ).

% bit_split_inv
tff(fact_816_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( aa_num_num(bit0,X22) = aa_num_num(bit0,Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_817_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_818_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_819_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_820_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_821_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_822_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_823_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_824_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_825_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y3: num] :
      ( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y3) )
    <=> ( X32 = Y3 ) ) ).

% verit_eq_simplify(9)
tff(fact_826_high__def,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% high_def
tff(fact_827_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_828_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_829_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(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(A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_830_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_831_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% psubsetD
tff(fact_832_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( aa_set_real_o(ord_less_set_real(A2),B2)
     => ( aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B2) ) ) ).

% psubsetD
tff(fact_833_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
     => ( aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% psubsetD
tff(fact_834_psubsetD,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ) ).

% psubsetD
tff(fact_835_less__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ord_less_fun_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2)) ) ).

% less_set_def
tff(fact_836_less__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B2)
    <=> ord_less_fun_real_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_set_def
tff(fact_837_less__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
    <=> ord_less_fun_int_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_set_def
tff(fact_838_less__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B2)
    <=> ord_le3534172602983796987VEBT_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_set_def
tff(fact_839_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M2),N)
             => aa_Extended_enat_o(P,M2) )
         => aa_Extended_enat_o(P,N) )
     => aa_Extended_enat_o(P,Na) ) ).

% enat_less_induct
tff(fact_840_power__divide,axiom,
    ! [A: complex,B: complex,Na: nat] : aa_nat_complex(power_power_complex(divide1717551699836669952omplex(A,B)),Na) = divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Na),aa_nat_complex(power_power_complex(B),Na)) ).

% power_divide
tff(fact_841_power__divide,axiom,
    ! [A: real,B: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(A),B)),Na) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ).

% power_divide
tff(fact_842_power__divide,axiom,
    ! [A: rat,B: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),Na) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ).

% power_divide
tff(fact_843_divide__numeral__1,axiom,
    ! [A: complex] : divide1717551699836669952omplex(A,numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_844_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_845_divide__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% divide_numeral_1
tff(fact_846_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_847_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_848_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_849_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_850_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_851_verit__comp__simplify1_I1_J,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),A) ).

% verit_comp_simplify1(1)
tff(fact_852_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_853_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_854_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_855_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_856_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_857_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_858_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_859_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_860_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_861_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_862_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_863_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_864_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_865_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.assoc
tff(fact_866_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_867_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_868_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_869_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_870_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_871_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_872_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_873_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_874_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_875_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_876_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_877_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_878_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_879_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_880_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_881_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_882_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_883_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_884_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_885_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_886_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_887_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_888_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_889_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_890_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% mult.assoc
tff(fact_891_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.assoc
tff(fact_892_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_893_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_894_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(A),B) = aa_real_real(times_times_real(B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_895_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_896_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_897_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_898_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(times_times_real(B),aa_real_real(times_times_real(A),C)) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_899_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_900_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_901_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_902_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).

% nat_int_comparison(1)
tff(fact_903_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,
        $ite((P),A,B)) = $ite((P),aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_if
tff(fact_904_field__sum__of__halves,axiom,
    ! [Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = Xa ).

% field_sum_of_halves
tff(fact_905_field__sum__of__halves,axiom,
    ! [Xa: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = Xa ).

% field_sum_of_halves
tff(fact_906_field__less__half__sum,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_907_field__less__half__sum,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_908_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_909_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_910_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_911_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_912_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),J)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J,L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_913_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_914_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_915_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_916_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_917_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I = J )
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J,L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_918_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_919_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_920_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_921_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_922_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),J)
        & ( K = L ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J,L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_923_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_924_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_strict_mono
tff(fact_925_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_926_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_927_add__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,D)) ) ) ).

% add_strict_mono
tff(fact_928_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_strict_left_mono
tff(fact_929_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_strict_left_mono
tff(fact_930_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_931_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_932_add__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(C,A)),plus_p5714425477246183910nteger(C,B)) ) ).

% add_strict_left_mono
tff(fact_933_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_strict_right_mono
tff(fact_934_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_strict_right_mono
tff(fact_935_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_936_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_937_add__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,C)) ) ).

% add_strict_right_mono
tff(fact_938_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_939_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_940_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_941_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_942_add__less__imp__less__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(C,A)),plus_p5714425477246183910nteger(C,B))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% add_less_imp_less_left
tff(fact_943_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_944_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_945_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_946_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_947_add__less__imp__less__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,C))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% add_less_imp_less_right
tff(fact_948_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != aa_num_num(bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_949_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : aa_num_num(bit0,X22) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_950_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_951_int__ops_I3_J,axiom,
    ! [Na: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% int_ops(3)
tff(fact_952_nat__numeral__as__int,axiom,
    ! [X5: num] : aa_num_nat(numeral_numeral_nat,X5) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X5)) ).

% nat_numeral_as_int
tff(fact_953_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_954_int__plus,axiom,
    ! [Na: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_955_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_956_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_957_nat__less__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_less_as_int
tff(fact_958_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_959_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,Na: nat] : divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ).

% div_exp_eq
tff(fact_960_div__exp__eq,axiom,
    ! [A: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ).

% div_exp_eq
tff(fact_961_div__exp__eq,axiom,
    ! [A: int,M: nat,Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ).

% div_exp_eq
tff(fact_962_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(divide1717551699836669952omplex(B,C)),A) = divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A),C) ).

% times_divide_eq_left
tff(fact_963_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(B),C)),A) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(B),A)),C) ).

% times_divide_eq_left
tff(fact_964_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),C) ).

% times_divide_eq_left
tff(fact_965_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : divide1717551699836669952omplex(divide1717551699836669952omplex(A,B),C) = divide1717551699836669952omplex(A,aa_complex_complex(times_times_complex(B),C)) ).

% divide_divide_eq_left
tff(fact_966_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(B),C)) ).

% divide_divide_eq_left
tff(fact_967_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% divide_divide_eq_left
tff(fact_968_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : divide1717551699836669952omplex(A,divide1717551699836669952omplex(B,C)) = divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C),B) ).

% divide_divide_eq_right
tff(fact_969_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(A),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),B) ).

% divide_divide_eq_right
tff(fact_970_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ).

% divide_divide_eq_right
tff(fact_971_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),divide1717551699836669952omplex(B,C)) = divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B),C) ).

% times_divide_eq_right
tff(fact_972_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),C) ).

% times_divide_eq_right
tff(fact_973_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) ).

% times_divide_eq_right
tff(fact_974_numeral__Bit1__div__2,axiom,
    ! [Na: num] : divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% numeral_Bit1_div_2
tff(fact_975_numeral__Bit1__div__2,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Na) ).

% numeral_Bit1_div_2
tff(fact_976_numeral__Bit1__div__2,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Na) ).

% numeral_Bit1_div_2
tff(fact_977_numeral__Bit0__div__2,axiom,
    ! [Na: num] : divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% numeral_Bit0_div_2
tff(fact_978_numeral__Bit0__div__2,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Na) ).

% numeral_Bit0_div_2
tff(fact_979_numeral__Bit0__div__2,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Na) ).

% numeral_Bit0_div_2
tff(fact_980_low__def,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% low_def
tff(fact_981_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_982_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mod_trivial
tff(fact_983_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_mod_trivial
tff(fact_984_real__divide__square__eq,axiom,
    ! [R: real,A: real] : aa_real_real(divide_divide_real(aa_real_real(times_times_real(R),A)),aa_real_real(times_times_real(R),R)) = aa_real_real(divide_divide_real(A),R) ).

% real_divide_square_eq
tff(fact_985_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_986_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self2
tff(fact_987_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_988_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_989_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self1
tff(fact_990_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_991_mod__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = M ) ) ).

% mod_less
tff(fact_992_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_993_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_994_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(B,C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_995_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_996_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_997_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(C,B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_998_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_999_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_1000_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,times_3573771949741848930nteger(B,C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_1001_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_1002_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_1003_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,times_3573771949741848930nteger(C,B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_1004_of__nat__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_mod
tff(fact_1005_of__nat__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_mod
tff(fact_1006_of__nat__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_mod
tff(fact_1007_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_1008_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_1009_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_right_eq
tff(fact_1010_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_1011_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_1012_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_left_eq
tff(fact_1013_mod__add__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_1014_mod__add__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_1015_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B4,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A3,B4),C) ) ) ) ).

% mod_add_cong
tff(fact_1016_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_1017_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_1018_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_eq
tff(fact_1019_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_1020_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_right_eq
tff(fact_1021_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(times_3573771949741848930nteger(A,B),C) ).

% mod_mult_right_eq
tff(fact_1022_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_1023_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_left_eq
tff(fact_1024_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(times_3573771949741848930nteger(A,B),C) ).

% mod_mult_left_eq
tff(fact_1025_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_1026_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ).

% mult_mod_right
tff(fact_1027_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : times_3573771949741848930nteger(C,modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(times_3573771949741848930nteger(C,A),times_3573771949741848930nteger(C,B)) ).

% mult_mod_right
tff(fact_1028_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_1029_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) ).

% mod_mult_mult2
tff(fact_1030_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(A,C),times_3573771949741848930nteger(B,C)) = times_3573771949741848930nteger(modulo364778990260209775nteger(A,B),C) ).

% mod_mult_mult2
tff(fact_1031_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),B4)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1032_mod__mult__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A3),B4)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1033_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B4,C) )
       => ( modulo364778990260209775nteger(times_3573771949741848930nteger(A,B),C) = modulo364778990260209775nteger(times_3573771949741848930nteger(A3,B4),C) ) ) ) ).

% mod_mult_cong
tff(fact_1034_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_1035_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_eq
tff(fact_1036_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(times_3573771949741848930nteger(A,B),C) ).

% mod_mult_eq
tff(fact_1037_power__mod,axiom,
    ! [A: nat,B: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),Na)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),B) ).

% power_mod
tff(fact_1038_power__mod,axiom,
    ! [A: int,B: int,Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),Na)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A),Na)),B) ).

% power_mod
tff(fact_1039_power__mod,axiom,
    ! [A: code_integer,B: code_integer,Na: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),Na),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na),B) ).

% power_mod
tff(fact_1040_real__of__nat__div__aux,axiom,
    ! [Xa: nat,D: nat] : aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),D))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_1041_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
     => ~ ! [D2: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ).

% mod_eqE
tff(fact_1042_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D2: code_integer] : B != plus_p5714425477246183910nteger(A,times_3573771949741848930nteger(C,D2)) ) ).

% mod_eqE
tff(fact_1043_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_1044_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_add1_eq
tff(fact_1045_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_1046_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_1047_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_mult1_eq
tff(fact_1048_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(times_3573771949741848930nteger(A,B),C) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(A,divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(times_3573771949741848930nteger(A,modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_1049_mod__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) ).

% mod_mult2_eq
tff(fact_1050_msrevs_I2_J,axiom,
    ! [K: nat,Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) ).

% msrevs(2)
tff(fact_1051_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Na)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_1052_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,Na)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_1053_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,Na: nat] : modulo364778990260209775nteger(A,times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_1054_div__mod__decomp,axiom,
    ! [A2: nat,Na: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),Na)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Na)) ).

% div_mod_decomp
tff(fact_1055_order__less__imp__not__less,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_1056_order__less__imp__not__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_1057_order__less__imp__not__less,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_1058_order__less__imp__not__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_1059_order__less__imp__not__less,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_1060_order__less__imp__not__less,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_1061_order__less__imp__not__eq2,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1062_order__less__imp__not__eq2,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1063_order__less__imp__not__eq2,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1064_order__less__imp__not__eq2,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1065_order__less__imp__not__eq2,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1066_order__less__imp__not__eq2,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1067_order__less__imp__not__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1068_order__less__imp__not__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1069_order__less__imp__not__eq,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1070_order__less__imp__not__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1071_order__less__imp__not__eq,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1072_order__less__imp__not__eq,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1073_linorder__less__linear,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
      | ( Xa = Y )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% linorder_less_linear
tff(fact_1074_linorder__less__linear,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
      | ( Xa = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% linorder_less_linear
tff(fact_1075_linorder__less__linear,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
      | ( Xa = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% linorder_less_linear
tff(fact_1076_linorder__less__linear,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
      | ( Xa = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% linorder_less_linear
tff(fact_1077_linorder__less__linear,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
      | ( Xa = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% linorder_less_linear
tff(fact_1078_linorder__less__linear,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
      | ( Xa = Y )
      | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ).

% linorder_less_linear
tff(fact_1079_order__less__imp__triv,axiom,
    ! [Xa: real,Y: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1080_order__less__imp__triv,axiom,
    ! [Xa: rat,Y: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1081_order__less__imp__triv,axiom,
    ! [Xa: num,Y: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1082_order__less__imp__triv,axiom,
    ! [Xa: nat,Y: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1083_order__less__imp__triv,axiom,
    ! [Xa: int,Y: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1084_order__less__imp__triv,axiom,
    ! [Xa: code_integer,Y: code_integer,P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1085_order__less__not__sym,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% order_less_not_sym
tff(fact_1086_order__less__not__sym,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% order_less_not_sym
tff(fact_1087_order__less__not__sym,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% order_less_not_sym
tff(fact_1088_order__less__not__sym,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% order_less_not_sym
tff(fact_1089_order__less__not__sym,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% order_less_not_sym
tff(fact_1090_order__less__not__sym,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ).

% order_less_not_sym
tff(fact_1091_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1092_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1093_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1094_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1095_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1096_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_re6978948838062957830nteger,C: code_integer] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,B)),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,X)),aa_real_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1097_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1098_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1099_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1100_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1101_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1102_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1103_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1104_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1105_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1106_order__less__subst1,axiom,
    ! [A: real,F: fun_Co3960896333048504454r_real,B: code_integer,C: code_integer] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_Code_integer_real(F,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => ( ! [X: code_integer,Y2: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Code_integer_real(F,X)),aa_Code_integer_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_Code_integer_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1107_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1108_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1109_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1110_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1111_order__less__irrefl,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Xa) ).

% order_less_irrefl
tff(fact_1112_order__less__irrefl,axiom,
    ! [Xa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Xa) ).

% order_less_irrefl
tff(fact_1113_order__less__irrefl,axiom,
    ! [Xa: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Xa) ).

% order_less_irrefl
tff(fact_1114_order__less__irrefl,axiom,
    ! [Xa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Xa) ).

% order_less_irrefl
tff(fact_1115_order__less__irrefl,axiom,
    ! [Xa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xa) ).

% order_less_irrefl
tff(fact_1116_order__less__irrefl,axiom,
    ! [Xa: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Xa) ).

% order_less_irrefl
tff(fact_1117_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1118_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1119_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1120_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1121_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1122_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_re6978948838062957830nteger,C: code_integer] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_Code_integer(F,B) = C )
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,X)),aa_real_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1123_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1124_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1125_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1126_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1127_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1128_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1129_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1130_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1131_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1132_ord__eq__less__subst,axiom,
    ! [A: code_integer,F: fun_re6978948838062957830nteger,B: real,C: real] :
      ( ( A = aa_real_Code_integer(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,X)),aa_real_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_real_Code_integer(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1133_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1134_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1135_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1136_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1137_order__less__trans,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1138_order__less__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1139_order__less__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1140_order__less__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1141_order__less__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1142_order__less__trans,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1143_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order_less_asym'
tff(fact_1144_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order_less_asym'
tff(fact_1145_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_1146_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order_less_asym'
tff(fact_1147_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order_less_asym'
tff(fact_1148_order__less__asym_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ).

% order_less_asym'
tff(fact_1149_linorder__neq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1150_linorder__neq__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1151_linorder__neq__iff,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1152_linorder__neq__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1153_linorder__neq__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1154_linorder__neq__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ( Xa != Y )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1155_order__less__asym,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% order_less_asym
tff(fact_1156_order__less__asym,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% order_less_asym
tff(fact_1157_order__less__asym,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% order_less_asym
tff(fact_1158_order__less__asym,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% order_less_asym
tff(fact_1159_order__less__asym,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% order_less_asym
tff(fact_1160_order__less__asym,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ).

% order_less_asym
tff(fact_1161_linorder__neqE,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_1162_linorder__neqE,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_1163_linorder__neqE,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_1164_linorder__neqE,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_1165_linorder__neqE,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_1166_linorder__neqE,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ( Xa != Y )
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_1167_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1168_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1169_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1170_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1171_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1172_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1173_linordered__field__no__ub,axiom,
    ! [X5: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X5),X_1) ).

% linordered_field_no_ub
tff(fact_1174_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),X_1) ).

% linordered_field_no_ub
tff(fact_1175_linordered__field__no__lb,axiom,
    ! [X5: real] :
    ? [Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X5) ).

% linordered_field_no_lb
tff(fact_1176_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X5) ).

% linordered_field_no_lb
tff(fact_1177_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1178_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1179_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1180_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1181_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1182_order_Ostrict__implies__not__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1183_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1184_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1185_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1186_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1187_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1188_dual__order_Ostrict__trans,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1189_not__less__iff__gr__or__eq,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1190_not__less__iff__gr__or__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1191_not__less__iff__gr__or__eq,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1192_not__less__iff__gr__or__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1193_not__less__iff__gr__or__eq,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1194_not__less__iff__gr__or__eq,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1195_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans
tff(fact_1196_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans
tff(fact_1197_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans
tff(fact_1198_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans
tff(fact_1199_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans
tff(fact_1200_order_Ostrict__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).

% order.strict_trans
tff(fact_1201_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,A: real,B: real] :
      ( ! [A4: real,B5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A4),B5)
         => aa_real_o(aa_real_fun_real_o(P,A4),B5) )
     => ( ! [A4: real] : aa_real_o(aa_real_fun_real_o(P,A4),A4)
       => ( ! [A4: real,B5: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B5),A4)
             => aa_real_o(aa_real_fun_real_o(P,A4),B5) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1202_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A4: rat,B5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B5)
         => aa_rat_o(aa_rat_fun_rat_o(P,A4),B5) )
     => ( ! [A4: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A4),A4)
       => ( ! [A4: rat,B5: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B5),A4)
             => aa_rat_o(aa_rat_fun_rat_o(P,A4),B5) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1203_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A4: num,B5: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B5)
         => aa_num_o(aa_num_fun_num_o(P,A4),B5) )
     => ( ! [A4: num] : aa_num_o(aa_num_fun_num_o(P,A4),A4)
       => ( ! [A4: num,B5: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B5),A4)
             => aa_num_o(aa_num_fun_num_o(P,A4),B5) )
         => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1204_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B5)
         => aa_nat_o(aa_nat_fun_nat_o(P,A4),B5) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),A4)
       => ( ! [A4: nat,B5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B5),A4)
             => aa_nat_o(aa_nat_fun_nat_o(P,A4),B5) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1205_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A4: int,B5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B5)
         => aa_int_o(aa_int_fun_int_o(P,A4),B5) )
     => ( ! [A4: int] : aa_int_o(aa_int_fun_int_o(P,A4),A4)
       => ( ! [A4: int,B5: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B5),A4)
             => aa_int_o(aa_int_fun_int_o(P,A4),B5) )
         => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1206_linorder__less__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,A: code_integer,B: code_integer] :
      ( ! [A4: code_integer,B5: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A4),B5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A4),B5) )
     => ( ! [A4: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A4),A4)
       => ( ! [A4: code_integer,B5: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,B5),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A4),B5) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1207_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [N2: nat] :
          ( aa_nat_o(P,N2)
          & ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
             => ~ aa_nat_o(P,M3) ) ) ) ).

% exists_least_iff
tff(fact_1208_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_1209_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_1210_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_1211_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_1212_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_1213_dual__order_Oirrefl,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),A) ).

% dual_order.irrefl
tff(fact_1214_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% dual_order.asym
tff(fact_1215_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% dual_order.asym
tff(fact_1216_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_1217_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_1218_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% dual_order.asym
tff(fact_1219_dual__order_Oasym,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% dual_order.asym
tff(fact_1220_linorder__cases,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( ( Xa != Y )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_cases
tff(fact_1221_linorder__cases,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( ( Xa != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_cases
tff(fact_1222_linorder__cases,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( ( Xa != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ) ).

% linorder_cases
tff(fact_1223_linorder__cases,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( ( Xa != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_cases
tff(fact_1224_linorder__cases,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( ( Xa != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_cases
tff(fact_1225_linorder__cases,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ( ( Xa != Y )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ) ).

% linorder_cases
tff(fact_1226_antisym__conv3,axiom,
    ! [Y: real,Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_1227_antisym__conv3,axiom,
    ! [Y: rat,Xa: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_1228_antisym__conv3,axiom,
    ! [Y: num,Xa: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_1229_antisym__conv3,axiom,
    ! [Y: nat,Xa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_1230_antisym__conv3,axiom,
    ! [Y: int,Xa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_1231_antisym__conv3,axiom,
    ! [Y: code_integer,Xa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_1232_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X)
             => aa_nat_o(P,Y4) )
         => aa_nat_o(P,X) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_1233_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( B = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1234_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1235_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1236_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1237_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1238_ord__less__eq__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( ( B = C )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1239_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = 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,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1240_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1241_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = 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,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1242_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1243_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1244_ord__eq__less__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A = B )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1245_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order.asym
tff(fact_1246_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order.asym
tff(fact_1247_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_1248_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order.asym
tff(fact_1249_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order.asym
tff(fact_1250_order_Oasym,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ).

% order.asym
tff(fact_1251_less__imp__neq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_1252_less__imp__neq,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_1253_less__imp__neq,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_1254_less__imp__neq,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_1255_less__imp__neq,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_1256_less__imp__neq,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_1257_dense,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Y) ) ) ).

% dense
tff(fact_1258_dense,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ? [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z3)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),Y) ) ) ).

% dense
tff(fact_1259_gt__ex,axiom,
    ! [Xa: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X_1) ).

% gt_ex
tff(fact_1260_gt__ex,axiom,
    ! [Xa: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X_1) ).

% gt_ex
tff(fact_1261_gt__ex,axiom,
    ! [Xa: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X_1) ).

% gt_ex
tff(fact_1262_gt__ex,axiom,
    ! [Xa: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X_1) ).

% gt_ex
tff(fact_1263_lt__ex,axiom,
    ! [Xa: real] :
    ? [Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),Xa) ).

% lt_ex
tff(fact_1264_lt__ex,axiom,
    ! [Xa: rat] :
    ? [Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),Xa) ).

% lt_ex
tff(fact_1265_lt__ex,axiom,
    ! [Xa: int] :
    ? [Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),Xa) ).

% lt_ex
tff(fact_1266_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1267_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1268_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) = aa_int_int(aa_int_fun_int_int(sup_sup_int,K),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1269_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: extended_enat,K: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( B2 = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,K),B) )
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B2) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,K),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1270_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: assn,K: assn,B: assn,A: assn] :
      ( ( B2 = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),B) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1271_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),A) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1272_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1273_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B) = aa_int_int(aa_int_fun_int_int(sup_sup_int,K),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1274_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: extended_enat,K: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( A2 = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,K),A) )
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A2),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,K),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1275_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: assn,K: assn,A: assn,B: assn] :
      ( ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),A) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1276_nat__div__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_div_as_int
tff(fact_1277_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% div_exp_mod_exp_eq
tff(fact_1278_div__exp__mod__exp__eq,axiom,
    ! [A: int,Na: nat,M: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(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,Na),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% div_exp_mod_exp_eq
tff(fact_1279_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Na: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) ).

% div_exp_mod_exp_eq
tff(fact_1280_bot_Onot__eq__extremum,axiom,
    ! [A: filter_nat] :
      ( ( A != bot_bot_filter_nat )
    <=> ord_less_filter_nat(bot_bot_filter_nat,A) ) ).

% bot.not_eq_extremum
tff(fact_1281_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> aa_set_real_o(ord_less_set_real(bot_bot_set_real),A) ) ).

% bot.not_eq_extremum
tff(fact_1282_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_set_nat(bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1283_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),A) ) ).

% bot.not_eq_extremum
tff(fact_1284_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1285_bot_Oextremum__strict,axiom,
    ! [A: filter_nat] : ~ ord_less_filter_nat(A,bot_bot_filter_nat) ).

% bot.extremum_strict
tff(fact_1286_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ aa_set_real_o(ord_less_set_real(A),bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_1287_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_1288_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ aa_set_int_o(ord_less_set_int(A),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_1289_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_1290_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : divide1717551699836669952omplex(plus_plus_complex(A,B),C) = plus_plus_complex(divide1717551699836669952omplex(A,C),divide1717551699836669952omplex(B,C)) ).

% add_divide_distrib
tff(fact_1291_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ).

% add_divide_distrib
tff(fact_1292_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) ).

% add_divide_distrib
tff(fact_1293_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : divide1717551699836669952omplex(divide1717551699836669952omplex(A,B),C) = divide1717551699836669952omplex(A,aa_complex_complex(times_times_complex(C),B)) ).

% divide_divide_eq_left'
tff(fact_1294_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(C),B)) ).

% divide_divide_eq_left'
tff(fact_1295_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ).

% divide_divide_eq_left'
tff(fact_1296_divide__divide__times__eq,axiom,
    ! [Xa: complex,Y: complex,Z: complex,W: complex] : divide1717551699836669952omplex(divide1717551699836669952omplex(Xa,Y),divide1717551699836669952omplex(Z,W)) = divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xa),W),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_1297_divide__divide__times__eq,axiom,
    ! [Xa: real,Y: real,Z: real,W: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(Xa),Y)),aa_real_real(divide_divide_real(Z),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xa),W)),aa_real_real(times_times_real(Y),Z)) ).

% divide_divide_times_eq
tff(fact_1298_divide__divide__times__eq,axiom,
    ! [Xa: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Z),W)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),W)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ).

% divide_divide_times_eq
tff(fact_1299_times__divide__times__eq,axiom,
    ! [Xa: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(divide1717551699836669952omplex(Xa,Y)),divide1717551699836669952omplex(Z,W)) = divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xa),Z),aa_complex_complex(times_times_complex(Y),W)) ).

% times_divide_times_eq
tff(fact_1300_times__divide__times__eq,axiom,
    ! [Xa: real,Y: real,Z: real,W: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(Xa),Y)),aa_real_real(divide_divide_real(Z),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xa),Z)),aa_real_real(times_times_real(Y),W)) ).

% times_divide_times_eq
tff(fact_1301_times__divide__times__eq,axiom,
    ! [Xa: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Z),W)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),W)) ).

% times_divide_times_eq
tff(fact_1302_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),bot_bot_assn) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_1303_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_real] : sup_sup_set_real(Xa,bot_bot_set_real) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_1304_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),bot_bot_set_nat) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_1305_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_int] : sup_sup_set_int(Xa,bot_bot_set_int) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_1306_div__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Q2) ).

% div_mult2_eq
tff(fact_1307_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% div_mult2_eq'
tff(fact_1308_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% div_mult2_eq'
tff(fact_1309_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,Na: nat] : divide6298287555418463151nteger(A,times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% div_mult2_eq'
tff(fact_1310_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1311_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1312_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = divide6298287555418463151nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1313_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_1314_div__mult2__numeral__eq,axiom,
    ! [A: code_integer,K: num,L: num] : divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,K)),aa_num_Code_integer(numera6620942414471956472nteger,L)) = divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_1315_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_1316_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_1317_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_1318_nat__mod__eq_H,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Na) = A ) ) ).

% nat_mod_eq'
tff(fact_1319_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_1320_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_1321_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% mult_div_mod_eq
tff(fact_1322_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(times_3573771949741848930nteger(B,divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_1323_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_1324_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_1325_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),times_3573771949741848930nteger(B,divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_1326_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1327_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1328_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),times_3573771949741848930nteger(divide6298287555418463151nteger(A,B),B)) = A ).

% mod_div_mult_eq
tff(fact_1329_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_1330_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% div_mult_mod_eq
tff(fact_1331_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_1332_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_1333_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_decomp
tff(fact_1334_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_1335_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1336_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1337_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_1338_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1339_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1340_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(B,divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_1341_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_1342_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,Na: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_1343_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,Na: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_1344_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_1345_zmod__helper,axiom,
    ! [Na: int,M: int,K: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),M) = K )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),A)),M) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A)),M) ) ) ).

% zmod_helper
tff(fact_1346_zmod__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_1347_div__mod__decomp__int,axiom,
    ! [A2: int,Na: int] : A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)),Na)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),Na)) ).

% div_mod_decomp_int
tff(fact_1348_pos__mod__bound2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% pos_mod_bound2
tff(fact_1349_linorder__neqE__linordered__idom,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1350_linorder__neqE__linordered__idom,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1351_linorder__neqE__linordered__idom,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1352_linorder__neqE__linordered__idom,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ( Xa != Y )
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1353_nat__mod__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_mod_as_int
tff(fact_1354_mod__exp__less__eq__exp,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% mod_exp_less_eq_exp
tff(fact_1355_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1356_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1357_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1358_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1359_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1360_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1361_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% comm_semiring_class.distrib
tff(fact_1362_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1363_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1364_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1365_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% distrib_left
tff(fact_1366_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% distrib_left
tff(fact_1367_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_1368_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_1369_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% distrib_right
tff(fact_1370_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% distrib_right
tff(fact_1371_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_1372_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_1373_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B),E)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E)),C) ).

% combine_common_factor
tff(fact_1374_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_1375_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_1376_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C) ).

% combine_common_factor
tff(fact_1377_nat__mod__eq,axiom,
    ! [B: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Na)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Na) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Na) = B ) ) ) ).

% nat_mod_eq
tff(fact_1378_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_1379_mod__plus__right,axiom,
    ! [A: nat,Xa: nat,M: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Xa)),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Xa)),M) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),M) ) ) ).

% mod_plus_right
tff(fact_1380_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,Na: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1381_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,Na: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1382_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,Na: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1383_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1384_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1385_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Na: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,one)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1386_nat__mod__eq__iff,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Na) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1387_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1388_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1389_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M)),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_1390_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Na: 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,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1391_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Na: 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,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1392_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Na: 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,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1393_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,Na: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_1394_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,Na: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_1395_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,Na: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_1396_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_1397_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,Na: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_1398_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,Na: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_1399_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),L)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),L)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),Na)),L) = Na ) ) ) ).

% word_rot_lem
tff(fact_1400_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N3)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N3)) ) ) ).

% mlex_bound
tff(fact_1401_mlex__fst__decrI,axiom,
    ! [A: nat,A3: nat,B: nat,N3: nat,B4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B4),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N3)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N3)),B4)) ) ) ) ).

% mlex_fst_decrI
tff(fact_1402_mlex__snd__decrI,axiom,
    ! [A: nat,A3: nat,B: nat,B4: nat,N3: nat] :
      ( ( A = A3 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N3)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N3)),B4)) ) ) ).

% mlex_snd_decrI
tff(fact_1403_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,A)),aa_nat_real(semiri5074537144036343181t_real,B)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_1404_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_1405_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,A)),aa_nat_nat(semiri1316708129612266289at_nat,B)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_1406_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,A)),aa_nat_rat(semiri681578069525770553at_rat,B)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_1407_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,B)) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_1408_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,A),aa_nat_complex(semiri8010041392384452111omplex,B)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_1409_sup__Un__eq,axiom,
    ! [R2: set_real,S: set_real,X5: real] :
      ( aa_real_o(sup_sup_fun_real_o(aTP_Lamm_a(R2),aTP_Lamm_a(S)),X5)
    <=> aa_set_real_o(member_real(X5),sup_sup_set_real(R2,S)) ) ).

% sup_Un_eq
tff(fact_1410_sup__Un__eq,axiom,
    ! [R2: set_int,S: set_int,X5: int] :
      ( aa_int_o(sup_sup_fun_int_o(aTP_Lamm_ac(R2),aTP_Lamm_ac(S)),X5)
    <=> aa_set_int_o(member_int(X5),sup_sup_set_int(R2,S)) ) ).

% sup_Un_eq
tff(fact_1411_sup__Un__eq,axiom,
    ! [R2: set_VEBT_VEBT,S: set_VEBT_VEBT,X5: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(sup_su3075289160232154067VEBT_o(aTP_Lamm_aa(R2),aTP_Lamm_aa(S)),X5)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),sup_su6272177626956685416T_VEBT(R2,S)) ) ).

% sup_Un_eq
tff(fact_1412_sup__Un__eq,axiom,
    ! [R2: set_nat,S: set_nat,X5: nat] :
      ( aa_nat_o(sup_sup_fun_nat_o(aTP_Lamm_ab(R2),aTP_Lamm_ab(S)),X5)
    <=> aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,R2),S)) ) ).

% sup_Un_eq
tff(fact_1413_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_less
tff(fact_1414_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_complex(power_power_complex(numera6690914467698888265omplex(K)),aa_num_nat(numeral_numeral_nat,L)) = numera6690914467698888265omplex(aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_1415_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_real(numeral_numeral_real,aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_1416_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_rat(numeral_numeral_rat,aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_1417_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_nat(numeral_numeral_nat,aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_1418_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_int(numeral_numeral_int,aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_1419_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_1420_the__elem__eq,axiom,
    ! [Xa: vEBT_VEBT] : the_elem_VEBT_VEBT(insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT)) = Xa ).

% the_elem_eq
tff(fact_1421_the__elem__eq,axiom,
    ! [Xa: real] : the_elem_real(insert_real(Xa,bot_bot_set_real)) = Xa ).

% the_elem_eq
tff(fact_1422_the__elem__eq,axiom,
    ! [Xa: nat] : the_elem_nat(insert_nat(Xa,bot_bot_set_nat)) = Xa ).

% the_elem_eq
tff(fact_1423_the__elem__eq,axiom,
    ! [Xa: int] : the_elem_int(insert_int(Xa,bot_bot_set_int)) = Xa ).

% the_elem_eq
tff(fact_1424_bot__empty__eq,axiom,
    ! [X5: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(bot_bo6217218057720408379VEBT_o,X5)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),bot_bo8194388402131092736T_VEBT) ) ).

% bot_empty_eq
tff(fact_1425_bot__empty__eq,axiom,
    ! [X5: real] :
      ( aa_real_o(bot_bot_fun_real_o,X5)
    <=> aa_set_real_o(member_real(X5),bot_bot_set_real) ) ).

% bot_empty_eq
tff(fact_1426_bot__empty__eq,axiom,
    ! [X5: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X5)
    <=> aa_set_nat_o(member_nat(X5),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_1427_bot__empty__eq,axiom,
    ! [X5: int] :
      ( aa_int_o(bot_bot_fun_int_o,X5)
    <=> aa_set_int_o(member_int(X5),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_1428_Collect__empty__eq__bot,axiom,
    ! [P: fun_complex_o] :
      ( ( collect_complex(P) = bot_bot_set_complex )
    <=> ( P = bot_bo8750099748576881071plex_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1429_Collect__empty__eq__bot,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( ( collec213857154873943460nt_int(P) = bot_bo1796632182523588997nt_int )
    <=> ( P = bot_bo3516217578726220278_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1430_Collect__empty__eq__bot,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ( P = bot_bot_fun_real_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1431_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1432_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1433_not__gr__zero,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_1434_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1435_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1436_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_1437_add__0,axiom,
    ! [A: complex] : plus_plus_complex(zero_zero_complex,A) = A ).

% add_0
tff(fact_1438_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_1439_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_1440_add__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% add_0
tff(fact_1441_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_1442_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) )
    <=> ( ( Xa = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_1443_add__eq__0__iff__both__eq__0,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = zero_zero_nat )
    <=> ( ( Xa = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_1444_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = plus_plus_complex(A,B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_1445_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_1446_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1447_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_1448_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_1449_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = plus_plus_complex(B,A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_1450_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_1451_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1452_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_1453_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_1454_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex(A,B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_1455_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_1456_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1457_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_1458_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_1459_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( plus_plus_complex(B,A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_1460_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_1461_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1462_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_1463_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_1464_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_1465_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1466_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_1467_add_Oright__neutral,axiom,
    ! [A: complex] : plus_plus_complex(A,zero_zero_complex) = A ).

% add.right_neutral
tff(fact_1468_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_1469_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_1470_add_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.right_neutral
tff(fact_1471_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_1472_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1473_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1474_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1475_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1476_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1477_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1478_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1479_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1480_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1481_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1482_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_1483_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_1484_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1485_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_1486_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_1487_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1488_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_1489_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1490_mult__zero__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_1491_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_1492_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_1493_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(times_times_real(zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_1494_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1495_mult__zero__left,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).

% mult_zero_left
tff(fact_1496_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_1497_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex(A,B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_1498_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_1499_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_1500_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( divide1717551699836669952omplex(C,A) = divide1717551699836669952omplex(C,B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1501_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(C),A) = aa_real_real(divide_divide_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1502_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1503_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( divide1717551699836669952omplex(A,C) = divide1717551699836669952omplex(B,C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1504_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1505_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1506_bits__div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_div_0
tff(fact_1507_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_1508_bits__div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_1509_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_1510_division__ring__divide__zero,axiom,
    ! [A: complex] : divide1717551699836669952omplex(A,zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_1511_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_1512_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_1513_div__0,axiom,
    ! [A: complex] : divide1717551699836669952omplex(zero_zero_complex,A) = zero_zero_complex ).

% div_0
tff(fact_1514_div__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_1515_div__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_1516_div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% div_0
tff(fact_1517_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1518_div__by__0,axiom,
    ! [A: complex] : divide1717551699836669952omplex(A,zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1519_div__by__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1520_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1521_div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_1522_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1523_bits__mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_mod_0
tff(fact_1524_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_1525_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_1526_mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% mod_0
tff(fact_1527_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_1528_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_1529_mod__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).

% mod_by_0
tff(fact_1530_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_1531_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_1532_mod__self,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).

% mod_self
tff(fact_1533_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_1534_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_1535_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_1536_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% neq0_conv
tff(fact_1537_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_1538_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_1539_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( Na = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_1540_mult__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1541_mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1542_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_1543_mult__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( Na = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1544_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_1545_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_1546_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1547_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1548_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1549_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1550_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1551_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1552_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1553_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1554_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_1555_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_1556_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_1557_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_1558_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),plus_p5714425477246183910nteger(B,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel2
tff(fact_1559_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_1560_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_1561_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_1562_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_1563_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),plus_p5714425477246183910nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel1
tff(fact_1564_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_1565_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_1566_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_1567_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_1568_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel2
tff(fact_1569_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_1570_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_1571_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_1572_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_1573_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(B,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel1
tff(fact_1574_sum__squares__eq__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xa),Xa)),aa_real_real(times_times_real(Y),Y)) = zero_zero_real )
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1575_sum__squares__eq__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1576_sum__squares__eq__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1577_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A),aa_complex_complex(times_times_complex(C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,divide1717551699836669952omplex(A,B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1578_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(divide_divide_real(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1579_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1580_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A),aa_complex_complex(times_times_complex(C),B)) = divide1717551699836669952omplex(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1581_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1582_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1583_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A),aa_complex_complex(times_times_complex(B),C)) = divide1717551699836669952omplex(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1584_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1585_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1586_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C),aa_complex_complex(times_times_complex(B),C)) = divide1717551699836669952omplex(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1587_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1588_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1589_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C),aa_complex_complex(times_times_complex(C),B)) = divide1717551699836669952omplex(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1590_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1591_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1592_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_1593_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_1594_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_1595_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_1596_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_1597_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% div_mult_mult1_if
tff(fact_1598_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1599_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1600_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1601_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1602_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1603_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1604_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1605_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1606_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1607_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1608_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_1609_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_1610_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_1611_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_1612_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,K)) = zero_z3403309356797280102nteger ).

% power_zero_numeral
tff(fact_1613_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_1614_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_1615_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_1616_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_1617_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_1618_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_1619_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(B,A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_1620_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1621_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1622_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1623_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1624_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = zero_z3403309356797280102nteger )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1625_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1626_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_1627_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_1628_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_1629_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_1630_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_1631_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_1632_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% semiring_1_class.of_nat_0
tff(fact_1633_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% semiring_1_class.of_nat_0
tff(fact_1634_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% semiring_1_class.of_nat_0
tff(fact_1635_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% semiring_1_class.of_nat_0
tff(fact_1636_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% semiring_1_class.of_nat_0
tff(fact_1637_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% semiring_1_class.of_nat_0
tff(fact_1638_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_1639_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_1640_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_1641_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_1642_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_1643_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_1644_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% add_gr_0
tff(fact_1645_div__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = zero_zero_nat ) ) ).

% div_less
tff(fact_1646_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1647_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_1648_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% mult_less_cancel2
tff(fact_1649_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: 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,Xa),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1650_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) ).

% nat_mult_div_cancel_disj
tff(fact_1651_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = divide1717551699836669952omplex(B,numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1652_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = 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(A),aa_num_real(numeral_numeral_real,W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1653_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1654_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( divide1717551699836669952omplex(B,numera6690914467698888265omplex(W)) = A )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1655_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B = aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1656_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1657_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_1658_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_1659_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_1660_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_1661_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_1662_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_1663_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_1664_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_1665_power__eq__0__iff,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_1666_power__eq__0__iff,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_nat_int(power_power_int(A),Na) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_1667_power__eq__0__iff,axiom,
    ! [A: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_1668_power__eq__0__iff,axiom,
    ! [A: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Na) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_1669_power__eq__0__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_1670_power__eq__0__iff,axiom,
    ! [A: rat,Na: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_1671_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),Na) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1672_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),Na) = M ) ) ).

% div_mult_self_is_m
tff(fact_1673_zero__less__ceiling,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% zero_less_ceiling
tff(fact_1674_zero__less__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa) ) ).

% zero_less_ceiling
tff(fact_1675_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_1676_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_1677_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1678_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_1679_zero__eq__power2,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
tff(fact_1680_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_1681_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_1682_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_1683_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_1684_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_1685_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_1686_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1687_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_1688_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_1689_zero__less__power2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
tff(fact_1690_sum__power2__eq__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1691_sum__power2__eq__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1692_sum__power2__eq__zero__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger )
    <=> ( ( Xa = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1693_sum__power2__eq__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1694_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: 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,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1695_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1696_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: 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,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1697_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: 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,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1698_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1699_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_1700_zero__reorient,axiom,
    ! [Xa: complex] :
      ( ( zero_zero_complex = Xa )
    <=> ( Xa = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_1701_zero__reorient,axiom,
    ! [Xa: real] :
      ( ( zero_zero_real = Xa )
    <=> ( Xa = zero_zero_real ) ) ).

% zero_reorient
tff(fact_1702_zero__reorient,axiom,
    ! [Xa: rat] :
      ( ( zero_zero_rat = Xa )
    <=> ( Xa = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1703_zero__reorient,axiom,
    ! [Xa: nat] :
      ( ( zero_zero_nat = Xa )
    <=> ( Xa = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1704_zero__reorient,axiom,
    ! [Xa: int] :
      ( ( zero_zero_int = Xa )
    <=> ( Xa = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1705_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1706_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,K) != zero_zero_real )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_1707_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,K) != zero_zero_int )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_1708_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,K) != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_1709_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,K) != zero_zero_rat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_1710_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,K) != zero_z3403309356797280102nteger )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_1711_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,K) != zero_zero_complex )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_1712_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = zero_zero_nat ) ) ).

% zero_power
tff(fact_1713_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(power_power_int(zero_zero_int),Na) = zero_zero_int ) ) ).

% zero_power
tff(fact_1714_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = zero_zero_real ) ) ).

% zero_power
tff(fact_1715_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Na) = zero_zero_complex ) ) ).

% zero_power
tff(fact_1716_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),Na) = zero_z3403309356797280102nteger ) ) ).

% zero_power
tff(fact_1717_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Na) = zero_zero_rat ) ) ).

% zero_power
tff(fact_1718_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_1719_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_1720_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_1721_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_1722_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_1723_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_1724_less__numeral__extra_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_numeral_extra(3)
tff(fact_1725_zero__less__iff__neq__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_1726_gr__implies__not__zero,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1727_not__less__zero,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less_zero
tff(fact_1728_gr__zeroI,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr_zeroI
tff(fact_1729_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_complex != numera6690914467698888265omplex(Na) ).

% zero_neq_numeral
tff(fact_1730_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Na) ).

% zero_neq_numeral
tff(fact_1731_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Na) ).

% zero_neq_numeral
tff(fact_1732_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Na) ).

% zero_neq_numeral
tff(fact_1733_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Na) ).

% zero_neq_numeral
tff(fact_1734_zero__neq__numeral,axiom,
    ! [Na: num] : zero_z3403309356797280102nteger != aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% zero_neq_numeral
tff(fact_1735_verit__sum__simplify,axiom,
    ! [A: complex] : plus_plus_complex(A,zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1736_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_1737_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_1738_verit__sum__simplify,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% verit_sum_simplify
tff(fact_1739_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_1740_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : plus_plus_complex(zero_zero_complex,A) = A ).

% add.group_left_neutral
tff(fact_1741_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_1742_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_1743_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_1744_add_Ocomm__neutral,axiom,
    ! [A: complex] : plus_plus_complex(A,zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_1745_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_1746_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_1747_add_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.comm_neutral
tff(fact_1748_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_1749_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : plus_plus_complex(zero_zero_complex,A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1750_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1751_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1752_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1753_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1754_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1755_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1756_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1757_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1758_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1759_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1760_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1761_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1762_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1763_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1764_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_1765_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(times_times_real(A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_1766_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1767_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_1768_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_1769_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_1770_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_1771_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1772_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_1773_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_1774_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_1775_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_1776_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1777_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_1778_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_1779_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: nat,Na: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1780_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,Na: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Na) != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1781_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: real,Na: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1782_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: complex,Na: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Na) != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1783_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: code_integer,Na: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1784_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,Na: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Na) != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1785_infinite__descent0,axiom,
    ! [P: fun_nat_o,Na: 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)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,Na) ) ) ).

% infinite_descent0
tff(fact_1786_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1787_less__zeroE,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_zeroE
tff(fact_1788_not__less0,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less0
tff(fact_1789_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1790_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr0I
tff(fact_1791_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_1792_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = M )
     => ( Na = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1793_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Na) = Na ).

% plus_nat.add_0
tff(fact_1794_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( K = zero_zero_nat )
        | ( M = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1795_mult__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Na) = zero_zero_nat ).

% mult_0
tff(fact_1796_lambda__zero,axiom,
    aTP_Lamm_by = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_1797_lambda__zero,axiom,
    aTP_Lamm_bz = times_times_real(zero_zero_real) ).

% lambda_zero
tff(fact_1798_lambda__zero,axiom,
    aTP_Lamm_ca = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_1799_lambda__zero,axiom,
    aTP_Lamm_cb = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_1800_lambda__zero,axiom,
    aTP_Lamm_cc = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_1801_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_1802_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_1803_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_1804_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_1805_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),zero_z3403309356797280102nteger) ).

% not_numeral_less_zero
tff(fact_1806_zero__less__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Na)) ).

% zero_less_numeral
tff(fact_1807_zero__less__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_less_numeral
tff(fact_1808_zero__less__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_less_numeral
tff(fact_1809_zero__less__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_less_numeral
tff(fact_1810_zero__less__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% zero_less_numeral
tff(fact_1811_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% pos_add_strict
tff(fact_1812_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% pos_add_strict
tff(fact_1813_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_1814_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_1815_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),plus_p5714425477246183910nteger(A,C)) ) ) ).

% pos_add_strict
tff(fact_1816_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C4: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) )
           => ( C4 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1817_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_1818_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_pos
tff(fact_1819_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_1820_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_1821_add__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,B)) ) ) ).

% add_pos_pos
tff(fact_1822_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_1823_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_1824_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_1825_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_1826_add__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_neg
tff(fact_1827_add__less__zeroD,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1828_add__less__zeroD,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1829_add__less__zeroD,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1830_add__less__zeroD,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(Xa,Y)),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),zero_z3403309356797280102nteger) ) ) ).

% add_less_zeroD
tff(fact_1831_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1832_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1833_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1834_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1835_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1836_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),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,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1837_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1838_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1839_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1840_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1841_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1842_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1843_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1844_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,C)) ) ) ).

% mult_strict_right_mono
tff(fact_1845_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1846_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1847_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1848_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1849_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),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,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1850_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1851_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1852_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1853_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1854_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1855_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1856_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1857_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B)) ) ) ).

% mult_strict_left_mono
tff(fact_1858_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1859_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1860_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1861_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1862_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1863_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1864_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1865_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1866_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1867_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1868_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1869_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1870_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1871_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1872_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1873_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1874_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),times_3573771949741848930nteger(B,A))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1875_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_1876_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1877_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1878_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_1879_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),times_3573771949741848930nteger(A,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos
tff(fact_1880_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_1881_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_1882_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_1883_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),times_3573771949741848930nteger(A,B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_less_mult_iff
tff(fact_1884_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_1885_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_1886_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_1887_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_1888_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(B,A)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg2
tff(fact_1889_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_pos_pos
tff(fact_1890_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1891_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1892_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1893_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),times_3573771949741848930nteger(A,B)) ) ) ).

% mult_pos_pos
tff(fact_1894_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_1895_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_1896_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_1897_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_1898_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,B)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg
tff(fact_1899_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_1900_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_1901_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_1902_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_1903_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,B)),zero_z3403309356797280102nteger) ) ) ).

% mult_neg_pos
tff(fact_1904_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1905_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1906_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1907_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1908_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_1909_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_1910_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_1911_not__square__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,A)),zero_z3403309356797280102nteger) ).

% not_square_less_zero
tff(fact_1912_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1913_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1914_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1915_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),times_3573771949741848930nteger(A,B)) ) ) ).

% mult_neg_neg
tff(fact_1916_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1917_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1918_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_1919_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_1920_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_1921_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_1922_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),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,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_1923_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_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,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_1924_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_1925_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_1926_divide__pos__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Xa),Y)) ) ) ).

% divide_pos_pos
tff(fact_1927_divide__pos__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) ) ) ).

% divide_pos_pos
tff(fact_1928_divide__pos__neg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xa),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_1929_divide__pos__neg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_1930_divide__neg__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xa),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_1931_divide__neg__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_1932_divide__neg__neg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Xa),Y)) ) ) ).

% divide_neg_neg
tff(fact_1933_divide__neg__neg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) ) ) ).

% divide_neg_neg
tff(fact_1934_zero__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_less_power
tff(fact_1935_zero__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_less_power
tff(fact_1936_zero__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_less_power
tff(fact_1937_zero__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_less_power
tff(fact_1938_zero__less__power,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ).

% zero_less_power
tff(fact_1939_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,Xa: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( divide1717551699836669952omplex(Xa,Y) = divide1717551699836669952omplex(W,Z) )
        <=> ( aa_complex_complex(times_times_complex(Xa),Z) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_1940_frac__eq__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(divide_divide_real(Xa),Y) = aa_real_real(divide_divide_real(W),Z) )
        <=> ( aa_real_real(times_times_real(Xa),Z) = aa_real_real(times_times_real(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_1941_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_1942_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( divide1717551699836669952omplex(B,C) = A )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_1943_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_1944_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_1945_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = divide1717551699836669952omplex(B,C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = B,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_1946_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_1947_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_1948_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( divide1717551699836669952omplex(B,C) = A ) ) ) ).

% divide_eq_imp
tff(fact_1949_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(times_times_real(A),C) )
       => ( aa_real_real(divide_divide_real(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_1950_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_1951_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = divide1717551699836669952omplex(B,C) ) ) ) ).

% eq_divide_imp
tff(fact_1952_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = B )
       => ( A = aa_real_real(divide_divide_real(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_1953_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B )
       => ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_1954_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex(B,C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1955_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(B),C) = A )
      <=> ( B = aa_real_real(times_times_real(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1956_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = A )
      <=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1957_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = divide1717551699836669952omplex(B,C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1958_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(divide_divide_real(B),C) )
      <=> ( aa_real_real(times_times_real(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1959_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1960_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_1961_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_1962_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_1963_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_1964_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_1965_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1966_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1967_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1968_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1969_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1970_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = A )
    <=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1971_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1972_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( Na = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1973_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_1974_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
      <=> ( M = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1975_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_1976_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_1977_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_1978_mod__less__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),Na) ) ).

% mod_less_divisor
tff(fact_1979_nat__mod__lem,axiom,
    ! [Na: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Na)
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Na) = B ) ) ) ).

% nat_mod_lem
tff(fact_1980_pow_Osimps_I1_J,axiom,
    ! [Xa: num] : aa_num_num(pow(Xa),one) = Xa ).

% pow.simps(1)
tff(fact_1981_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_1982_sum__squares__gt__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xa),Xa)),aa_real_real(times_times_real(Y),Y)))
    <=> ( ( Xa != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1983_sum__squares__gt__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))
    <=> ( ( Xa != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1984_sum__squares__gt__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( Xa != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1985_sum__squares__gt__zero__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(times_3573771949741848930nteger(Xa,Xa),times_3573771949741848930nteger(Y,Y)))
    <=> ( ( Xa != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1986_not__sum__squares__lt__zero,axiom,
    ! [Xa: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xa),Xa)),aa_real_real(times_times_real(Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_1987_not__sum__squares__lt__zero,axiom,
    ! [Xa: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_1988_not__sum__squares__lt__zero,axiom,
    ! [Xa: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_1989_not__sum__squares__lt__zero,axiom,
    ! [Xa: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(times_3573771949741848930nteger(Xa,Xa),times_3573771949741848930nteger(Y,Y))),zero_z3403309356797280102nteger) ).

% not_sum_squares_lt_zero
tff(fact_1990_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1991_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1992_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_1993_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_1994_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Z),Y)),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(divide_divide_real(Xa),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_1995_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_1996_mult__imp__div__pos__less,axiom,
    ! [Y: real,Xa: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(times_times_real(Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xa),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_1997_mult__imp__div__pos__less,axiom,
    ! [Y: rat,Xa: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_1998_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_1999_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2000_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2001_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2002_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2003_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2004_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2005_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2006_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(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(A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_2007_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_2008_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_2009_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_2010_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = divide1717551699836669952omplex(B,C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2011_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_2012_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_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_2013_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( divide1717551699836669952omplex(B,C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2014_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_2015_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_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_2016_divide__add__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( plus_plus_complex(divide1717551699836669952omplex(Xa,Z),Y) = divide1717551699836669952omplex(plus_plus_complex(Xa,aa_complex_complex(times_times_complex(Y),Z)),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2017_divide__add__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(Xa),Z)),Y) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2018_divide__add__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2019_add__divide__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( plus_plus_complex(Xa,divide1717551699836669952omplex(Y,Z)) = divide1717551699836669952omplex(plus_plus_complex(aa_complex_complex(times_times_complex(Xa),Z),Y),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2020_add__divide__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(divide_divide_real(Y),Z)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xa),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2021_add__divide__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2022_add__num__frac,axiom,
    ! [Y: complex,Z: complex,Xa: complex] :
      ( ( Y != zero_zero_complex )
     => ( plus_plus_complex(Z,divide1717551699836669952omplex(Xa,Y)) = divide1717551699836669952omplex(plus_plus_complex(Xa,aa_complex_complex(times_times_complex(Z),Y)),Y) ) ) ).

% add_num_frac
tff(fact_2023_add__num__frac,axiom,
    ! [Y: real,Z: real,Xa: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(divide_divide_real(Xa),Y)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2024_add__num__frac,axiom,
    ! [Y: rat,Z: rat,Xa: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2025_add__frac__num,axiom,
    ! [Y: complex,Xa: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( plus_plus_complex(divide1717551699836669952omplex(Xa,Y),Z) = divide1717551699836669952omplex(plus_plus_complex(Xa,aa_complex_complex(times_times_complex(Z),Y)),Y) ) ) ).

% add_frac_num
tff(fact_2026_add__frac__num,axiom,
    ! [Y: real,Xa: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(Xa),Y)),Z) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2027_add__frac__num,axiom,
    ! [Y: rat,Xa: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2028_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xa: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( plus_plus_complex(divide1717551699836669952omplex(Xa,Y),divide1717551699836669952omplex(W,Z)) = divide1717551699836669952omplex(plus_plus_complex(aa_complex_complex(times_times_complex(Xa),Z),aa_complex_complex(times_times_complex(W),Y)),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2029_add__frac__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(Xa),Y)),aa_real_real(divide_divide_real(W),Z)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xa),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2030_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2031_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      plus_plus_complex(A,divide1717551699836669952omplex(B,Z)) = $ite(Z = zero_zero_complex,A,divide1717551699836669952omplex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z),B),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2032_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(divide_divide_real(B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2033_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2034_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      plus_plus_complex(divide1717551699836669952omplex(A,Z),B) = $ite(Z = zero_zero_complex,B,divide1717551699836669952omplex(plus_plus_complex(A,aa_complex_complex(times_times_complex(B),Z)),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2035_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2036_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2037_memb__imp__not__empty,axiom,
    ! [Xa: vEBT_VEBT,S: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),S)
     => ( S != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
tff(fact_2038_memb__imp__not__empty,axiom,
    ! [Xa: real,S: set_real] :
      ( aa_set_real_o(member_real(Xa),S)
     => ( S != bot_bot_set_real ) ) ).

% memb_imp_not_empty
tff(fact_2039_memb__imp__not__empty,axiom,
    ! [Xa: nat,S: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),S)
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_2040_memb__imp__not__empty,axiom,
    ! [Xa: int,S: set_int] :
      ( aa_set_int_o(member_int(Xa),S)
     => ( S != bot_bot_set_int ) ) ).

% memb_imp_not_empty
tff(fact_2041_set__notEmptyE,axiom,
    ! [S: set_VEBT_VEBT] :
      ( ( S != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S) ) ).

% set_notEmptyE
tff(fact_2042_set__notEmptyE,axiom,
    ! [S: set_real] :
      ( ( S != bot_bot_set_real )
     => ~ ! [X: real] : ~ aa_set_real_o(member_real(X),S) ) ).

% set_notEmptyE
tff(fact_2043_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X: nat] : ~ aa_set_nat_o(member_nat(X),S) ) ).

% set_notEmptyE
tff(fact_2044_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X: int] : ~ aa_set_int_o(member_int(X),S) ) ).

% set_notEmptyE
tff(fact_2045_ex__less__of__nat__mult,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),Xa)) ) ).

% ex_less_of_nat_mult
tff(fact_2046_ex__less__of__nat__mult,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),Xa)) ) ).

% ex_less_of_nat_mult
tff(fact_2047_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),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,Na)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2048_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),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,Na)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2049_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q2: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2050_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_2051_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_2052_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_2053_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q2)),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_2054_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_2055_td__gal__lt,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),B) ) ) ).

% td_gal_lt
tff(fact_2056_mod__plus__cong,axiom,
    ! [B: int,B4: int,Xa: int,X6: int,Y: int,Y5: int,Z4: int] :
      ( ( B = B4 )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),B4) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X6),B4) )
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y),B4) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y5),B4) )
         => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),Y5) = Z4 )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Z4),B4) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_2057_div__less__mono,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Na) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Na) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_2058_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_2059_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(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_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(aa_rat_fun_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_2060_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_2061_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),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(aa_rat_fun_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(aa_rat_fun_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_2062_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_2063_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_2064_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_2065_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_2066_zero__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% zero_power2
tff(fact_2067_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_2068_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_2069_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_2070_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q2: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_2071_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_2072_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)))) ) ).

% dividend_less_times_div
tff(fact_2073_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_2074_split__div,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Na != zero_zero_nat )
         => ! [I2: nat,J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I2)),J2) )
               => aa_nat_o(P,I2) ) ) ) ) ) ).

% split_div
tff(fact_2075_msrevs_I1_J,axiom,
    ! [Na: nat,K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),K) ) ) ).

% msrevs(1)
tff(fact_2076_split__mod,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,M) )
        & ( ( Na != zero_zero_nat )
         => ! [I2: nat,J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I2)),J2) )
               => aa_nat_o(P,J2) ) ) ) ) ) ).

% split_mod
tff(fact_2077_mod__lemma,axiom,
    ! [C: nat,R: nat,B: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Q2),C))),R)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mod_lemma
tff(fact_2078_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_2079_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_2080_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_2081_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2082_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2083_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_2084_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_2085_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_2086_power2__less__0,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger) ).

% power2_less_0
tff(fact_2087_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != 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))),Na) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2088_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2089_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Na: nat] :
      ( ( 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,M),Na)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2090_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2091_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2092_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Na: nat] :
      ( ( 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,M),Na)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2093_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Na: 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,Na))),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,Na)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2094_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Na: 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,Na))),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,Na)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2095_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Na: 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,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2096_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2097_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2098_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M)),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,M),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2099_sum__power2__gt__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xa != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2100_sum__power2__gt__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xa != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2101_sum__power2__gt__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xa != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2102_sum__power2__gt__zero__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xa != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2103_not__sum__power2__lt__zero,axiom,
    ! [Xa: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_2104_not__sum__power2__lt__zero,axiom,
    ! [Xa: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_2105_not__sum__power2__lt__zero,axiom,
    ! [Xa: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_2106_not__sum__power2__lt__zero,axiom,
    ! [Xa: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger) ).

% not_sum_power2_lt_zero
tff(fact_2107_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2108_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2109_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B))),B)
       => ( modulo364778990260209775nteger(A,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2110_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_2111_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_2112_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_2113_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2114_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2115_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B))),B)
       => ( times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),divide6298287555418463151nteger(A,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2116_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2117_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_2118_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_2119_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_2120_unset__bit__0,axiom,
    ! [A: code_integer] : bit_se8260200283734997820nteger(zero_zero_nat,A) = times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_2121_unset__bit__0,axiom,
    ! [A: int] : bit_se4203085406695923979it_int(zero_zero_nat,A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_2122_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_2123_T__vebt__buildupi__gq__0,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% T_vebt_buildupi_gq_0
tff(fact_2124_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Na: nat] :
      ( ! [M4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M4),zero_zero_nat)
     => ( ! [M4: 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,M4),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M4),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,M),Na) ) ) ).

% gcd_nat_induct
tff(fact_2125_add__scale__eq__noteq,axiom,
    ! [R: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( plus_plus_complex(A,aa_complex_complex(times_times_complex(R),C)) != plus_plus_complex(B,aa_complex_complex(times_times_complex(R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2126_add__scale__eq__noteq,axiom,
    ! [R: real,A: real,B: real,C: real,D: real] :
      ( ( R != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(times_times_real(R),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(times_times_real(R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2127_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2128_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2129_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2130_mult__less__iff1,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Xa),Z)),aa_real_real(times_times_real(Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_2131_mult__less__iff1,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_2132_mult__less__iff1,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_2133_mult__less__iff1,axiom,
    ! [Z: code_integer,Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(Xa,Z)),times_3573771949741848930nteger(Y,Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_2134_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_2135_arith__geo__mean,axiom,
    ! [U: real,Xa: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(Xa),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2136_arith__geo__mean,axiom,
    ! [U: rat,Xa: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2137_min__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xa)
    <=> ( aa_set_nat_o(member_nat(Xa),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4) ) ) ) ).

% min_in_set_def
tff(fact_2138_max__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xa)
    <=> ( aa_set_nat_o(member_nat(Xa),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa) ) ) ) ).

% max_in_set_def
tff(fact_2139_order__refl,axiom,
    ! [Xa: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Xa),Xa) ).

% order_refl
tff(fact_2140_order__refl,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Xa) ).

% order_refl
tff(fact_2141_order__refl,axiom,
    ! [Xa: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Xa) ).

% order_refl
tff(fact_2142_order__refl,axiom,
    ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Xa) ).

% order_refl
tff(fact_2143_order__refl,axiom,
    ! [Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xa) ).

% order_refl
tff(fact_2144_dual__order_Orefl,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% dual_order.refl
tff(fact_2145_dual__order_Orefl,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% dual_order.refl
tff(fact_2146_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_2147_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_2148_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_2149_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(71)
tff(fact_2150_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Na) ).

% semiring_norm(68)
tff(fact_2151_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_2152_le0,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% le0
tff(fact_2153_subset__empty,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real)
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_2154_subset__empty,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_2155_subset__empty,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_2156_empty__subsetI,axiom,
    ! [A2: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A2) ).

% empty_subsetI
tff(fact_2157_empty__subsetI,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A2) ).

% empty_subsetI
tff(fact_2158_empty__subsetI,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A2) ).

% empty_subsetI
tff(fact_2159_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(73)
tff(fact_2160_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_2161_insert__subset,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(insert_real(Xa,A2)),B2)
    <=> ( aa_set_real_o(member_real(Xa),B2)
        & aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ) ).

% insert_subset
tff(fact_2162_insert__subset,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(insert_int(Xa,A2)),B2)
    <=> ( aa_set_int_o(member_int(Xa),B2)
        & aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ) ).

% insert_subset
tff(fact_2163_insert__subset,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(Xa,A2)),B2)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B2)
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2) ) ) ).

% insert_subset
tff(fact_2164_insert__subset,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(insert_nat(Xa,A2)),B2)
    <=> ( aa_set_nat_o(member_nat(Xa),B2)
        & aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ) ).

% insert_subset
tff(fact_2165_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se4203085406695923979it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2166_Un__subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),C2)
        & aa_set_nat_o(ord_less_eq_set_nat(B2),C2) ) ) ).

% Un_subset_iff
tff(fact_2167_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( ( A2 != B2 )
       => aa_set_nat_o(ord_less_set_nat(A2),B2) ) ) ).

% psubsetI
tff(fact_2168_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na)
    <=> ( Na != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_2169_T__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% T_vebt_buildupi_univ
tff(fact_2170_le__zero__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_2171_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2172_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2173_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2174_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2175_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2176_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_2177_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_2178_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_2179_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_2180_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_2181_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_2182_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_2183_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_2184_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),one) ).

% semiring_norm(69)
tff(fact_2185_mult_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% mult.right_neutral
tff(fact_2186_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.right_neutral
tff(fact_2187_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_2188_mult_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.right_neutral
tff(fact_2189_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.right_neutral
tff(fact_2190_mult__1,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% mult_1
tff(fact_2191_mult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% mult_1
tff(fact_2192_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% mult_1
tff(fact_2193_mult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% mult_1
tff(fact_2194_mult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% mult_1
tff(fact_2195_div__by__1,axiom,
    ! [A: complex] : divide1717551699836669952omplex(A,one_one_complex) = A ).

% div_by_1
tff(fact_2196_div__by__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),one_one_real) = A ).

% div_by_1
tff(fact_2197_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),one_one_rat) = A ).

% div_by_1
tff(fact_2198_div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% div_by_1
tff(fact_2199_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_2200_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_2201_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_2202_power__one,axiom,
    ! [Na: nat] : aa_nat_assn(power_power_assn(one_one_assn),Na) = one_one_assn ).

% power_one
tff(fact_2203_power__one,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Na) = one_one_nat ).

% power_one
tff(fact_2204_power__one,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(one_one_int),Na) = one_one_int ).

% power_one
tff(fact_2205_power__one,axiom,
    ! [Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Na) = one_one_real ).

% power_one
tff(fact_2206_power__one,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(one_one_complex),Na) = one_one_complex ).

% power_one
tff(fact_2207_power__one,axiom,
    ! [Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),Na) = one_one_Code_integer ).

% power_one
tff(fact_2208_power__one,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(one_one_rat),Na) = one_one_rat ).

% power_one
tff(fact_2209_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(72)
tff(fact_2210_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_2211_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_2212_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_2213_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_2214_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_2215_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_2216_le__sup__iff,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)),Z)
    <=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),Z)
        & aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y),Z) ) ) ).

% le_sup_iff
tff(fact_2217_le__sup__iff,axiom,
    ! [Xa: assn,Y: assn,Z: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y),Z)
    <=> ( ord_less_eq_assn(Xa,Z)
        & ord_less_eq_assn(Y,Z) ) ) ).

% le_sup_iff
tff(fact_2218_le__sup__iff,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)),Z)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Z)
        & aa_set_nat_o(ord_less_eq_set_nat(Y),Z) ) ) ).

% le_sup_iff
tff(fact_2219_le__sup__iff,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_2220_le__sup__iff,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_2221_le__sup__iff,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) ) ) ).

% le_sup_iff
tff(fact_2222_sup_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,B),C)),A)
    <=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
        & aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A) ) ) ).

% sup.bounded_iff
tff(fact_2223_sup_Obounded__iff,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C),A)
    <=> ( ord_less_eq_assn(B,A)
        & ord_less_eq_assn(C,A) ) ) ).

% sup.bounded_iff
tff(fact_2224_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & aa_set_nat_o(ord_less_eq_set_nat(C),A) ) ) ).

% sup.bounded_iff
tff(fact_2225_sup_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% sup.bounded_iff
tff(fact_2226_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% sup.bounded_iff
tff(fact_2227_sup_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% sup.bounded_iff
tff(fact_2228_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_2229_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_2230_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_2231_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_2232_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_2233_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_2234_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2235_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2236_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2237_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2238_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2239_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2240_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Na) = one_one_real )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2241_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Na) = one_one_int )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2242_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Na) = one_one_nat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2243_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Na) = one_one_rat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2244_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Na) = one_one_Code_integer )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2245_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Na) = one_one_complex )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2246_singleton__insert__inj__eq,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT) = insert_VEBT_VEBT(A,A2) )
    <=> ( ( A = B )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq
tff(fact_2247_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( insert_real(B,bot_bot_set_real) = insert_real(A,A2) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq
tff(fact_2248_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A2) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_2249_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A2) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_2250_singleton__insert__inj__eq_H,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( insert_VEBT_VEBT(A,A2) = insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT) )
    <=> ( ( A = B )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2251_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( insert_real(A,A2) = insert_real(B,bot_bot_set_real) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2252_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( insert_int(A,A2) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2253_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( insert_nat(A,A2) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2254_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se4203085406695923979it_int(Na,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_2255_not__real__square__gt__zero,axiom,
    ! [Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(Xa),Xa))
    <=> ( Xa = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_2256_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(1)
tff(fact_2257_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2258_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2259_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2260_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2261_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2262_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2263_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_2264_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_2265_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_2266_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_2267_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_2268_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_2269_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_2270_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_2271_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_2272_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_2273_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_2274_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_2275_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_2276_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_2277_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_2278_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_2279_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_2280_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_2281_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_2282_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_2283_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_2284_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(times_times_real(C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2285_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_2286_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_2287_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_2288_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_2289_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_2290_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_2291_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2292_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(times_times_real(A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2293_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2294_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_2295_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numera6690914467698888265omplex(Na) = one_one_complex )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2296_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_real(numeral_numeral_real,Na) = one_one_real )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2297_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Na) = one_one_rat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2298_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Na) = one_one_nat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2299_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_int(numeral_numeral_int,Na) = one_one_int )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2300_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_Code_integer(numera6620942414471956472nteger,Na) = one_one_Code_integer )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2301_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2302_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2303_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2304_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2305_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2306_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_Code_integer = aa_num_Code_integer(numera6620942414471956472nteger,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2307_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( divide1717551699836669952omplex(A,A) = one_one_complex ) ) ).

% div_self
tff(fact_2308_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% div_self
tff(fact_2309_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% div_self
tff(fact_2310_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_2311_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_2312_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex(A,B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2313_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2314_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2315_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = divide1717551699836669952omplex(A,B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2316_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2317_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2318_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( divide1717551699836669952omplex(A,A) = one_one_complex ) ) ).

% divide_self
tff(fact_2319_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2320_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2321_divide__self__if,axiom,
    ! [A: complex] :
      divide1717551699836669952omplex(A,A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_2322_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(divide_divide_real(A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_2323_divide__self__if,axiom,
    ! [A: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_2324_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2325_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2326_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2327_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2328_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(divide_divide_real(one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2329_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2330_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(divide_divide_real(one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2331_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2332_power__inject__exp,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_2333_power__inject__exp,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_2334_power__inject__exp,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_2335_power__inject__exp,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_2336_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),M) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_2337_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_2338_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_2339_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_2340_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_2341_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_2342_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_2343_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% mult_le_cancel2
tff(fact_2344_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2345_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_2346_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_2347_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),aa_nat_real(semiri5074537144036343181t_real,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_2348_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_2349_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_2350_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_2351_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_2352_ceiling__one,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_2353_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_2354_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_2355_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_2356_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(79)
tff(fact_2357_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(74)
tff(fact_2358_nat__ceiling__le__eq,axiom,
    ! [Xa: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xa))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_2359_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2360_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),one_one_Code_integer)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2361_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2362_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2363_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2364_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_2365_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_2366_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_2367_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_2368_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_2369_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_2370_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2371_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2372_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2373_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2374_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2375_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2376_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2377_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2378_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_2379_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_2380_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2381_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2382_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(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(A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2383_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2384_power__increasing__iff,axiom,
    ! [B: real,Xa: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_2385_power__increasing__iff,axiom,
    ! [B: code_integer,Xa: nat,Y: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_2386_power__increasing__iff,axiom,
    ! [B: rat,Xa: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Xa)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_2387_power__increasing__iff,axiom,
    ! [B: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_2388_power__increasing__iff,axiom,
    ! [B: int,Xa: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Xa)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_2389_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( divide1717551699836669952omplex(B,aa_complex_complex(times_times_complex(A),B)) = divide1717551699836669952omplex(one_one_complex,A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2390_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(B),aa_real_real(times_times_real(A),B)) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2391_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2392_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( divide1717551699836669952omplex(A,aa_complex_complex(times_times_complex(A),B)) = divide1717551699836669952omplex(one_one_complex,B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2393_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(A),B)) = aa_real_real(divide_divide_real(one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2394_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2395_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_2396_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_2397_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_2398_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_2399_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_2400_power__strict__increasing__iff,axiom,
    ! [B: real,Xa: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2401_power__strict__increasing__iff,axiom,
    ! [B: rat,Xa: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Xa)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2402_power__strict__increasing__iff,axiom,
    ! [B: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2403_power__strict__increasing__iff,axiom,
    ! [B: int,Xa: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Xa)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2404_power__strict__increasing__iff,axiom,
    ! [B: code_integer,Xa: nat,Y: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2405_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: 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,Xa))
    <=> 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)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2406_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: 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,Xa))
    <=> 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)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2407_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: 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,Xa))
    <=> 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)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2408_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: 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,Xa))
    <=> 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)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2409_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> 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)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2410_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2411_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2412_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2413_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2414_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2415_ceiling__le__zero,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_2416_ceiling__le__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_2417_ceiling__le__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_num_real(numeral_numeral_real,V)) ) ).

% ceiling_le_numeral
tff(fact_2418_ceiling__le__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_2419_one__le__ceiling,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% one_le_ceiling
tff(fact_2420_one__le__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa) ) ).

% one_le_ceiling
tff(fact_2421_ceiling__le__one,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ).

% ceiling_le_one
tff(fact_2422_ceiling__le__one,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),one_one_rat) ) ).

% ceiling_le_one
tff(fact_2423_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_2424_int__div__same__is__1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = A )
      <=> ( B = one_one_int ) ) ) ).

% int_div_same_is_1
tff(fact_2425_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2426_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2427_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2428_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2429_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2430_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2431_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2432_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2433_power__decreasing__iff,axiom,
    ! [B: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2434_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2435_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2436_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2437_power__decreasing__iff,axiom,
    ! [B: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2438_one__add__one,axiom,
    plus_plus_complex(one_one_complex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2439_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2440_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_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_2441_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_2442_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_2443_one__add__one,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2444_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2445_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2446_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2447_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2448_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2449_power__mono__iff,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2450_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na))
          <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2451_power__mono__iff,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2452_power__mono__iff,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2453_power__mono__iff,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2454_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_2455_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_2456_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_2457_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_2458_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_2459_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_2460_numeral__plus__one,axiom,
    ! [Na: num] : plus_plus_complex(numera6690914467698888265omplex(Na),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_2461_numeral__plus__one,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_2462_numeral__plus__one,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_2463_numeral__plus__one,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_2464_numeral__plus__one,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_2465_numeral__plus__one,axiom,
    ! [Na: num] : plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_2466_one__plus__numeral,axiom,
    ! [Na: num] : plus_plus_complex(one_one_complex,numera6690914467698888265omplex(Na)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_2467_one__plus__numeral,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_2468_one__plus__numeral,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_2469_one__plus__numeral,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_2470_one__plus__numeral,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_2471_one__plus__numeral,axiom,
    ! [Na: num] : plus_p5714425477246183910nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_2472_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_2473_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_2474_ceiling__less__one,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_2475_ceiling__less__one,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_2476_one__less__ceiling,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ).

% one_less_ceiling
tff(fact_2477_one__less__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xa) ) ).

% one_less_ceiling
tff(fact_2478_ceiling__add__one,axiom,
    ! [Xa: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xa)),one_one_int) ).

% ceiling_add_one
tff(fact_2479_ceiling__add__one,axiom,
    ! [Xa: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),one_one_int) ).

% ceiling_add_one
tff(fact_2480_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2481_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2482_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2483_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2484_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2485_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_2486_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2487_bits__1__div__2,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_2488_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_2489_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_2490_one__div__two__eq__zero,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_2491_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_2492_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_2493_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_2494_power2__less__eq__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
tff(fact_2495_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_2496_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_2497_power2__eq__iff__nonneg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2498_power2__eq__iff__nonneg,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2499_power2__eq__iff__nonneg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2500_power2__eq__iff__nonneg,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2501_power2__eq__iff__nonneg,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2502_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: 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)),Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2503_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Na)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2504_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: 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)),Na)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2505_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: 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)),Na)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2506_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> 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)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2507_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2508_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2509_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2510_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2511_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2512_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_2513_one__mod__exp__eq__one,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) = one_one_int ).

% one_mod_exp_eq_one
tff(fact_2514_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_2515_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_2516_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_2517_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_2518_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_2519_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_2520_le__cases3,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xa) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xa)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2521_le__cases3,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xa) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xa)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2522_le__cases3,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xa) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xa)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2523_le__cases3,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xa) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xa)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2524_order__class_Oorder__eq__iff,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( Xa = Y )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
        & aa_set_nat_o(ord_less_eq_set_nat(Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_2525_order__class_Oorder__eq__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa = Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_2526_order__class_Oorder__eq__iff,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa = Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_2527_order__class_Oorder__eq__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_2528_order__class_Oorder__eq__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_2529_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2530_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2531_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = 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,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2532_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2533_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2534_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( ( B = C )
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2535_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2536_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2537_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2538_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2539_order__antisym,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_2540_order__antisym,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_2541_order__antisym,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_2542_order__antisym,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_2543_order__antisym,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_2544_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% order.trans
tff(fact_2545_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% order.trans
tff(fact_2546_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% order.trans
tff(fact_2547_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% order.trans
tff(fact_2548_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% order.trans
tff(fact_2549_order__trans,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Y),Z)
       => aa_set_nat_o(ord_less_eq_set_nat(Xa),Z) ) ) ).

% order_trans
tff(fact_2550_order__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z) ) ) ).

% order_trans
tff(fact_2551_order__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Z) ) ) ).

% order_trans
tff(fact_2552_order__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z) ) ) ).

% order_trans
tff(fact_2553_order__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z) ) ) ).

% order_trans
tff(fact_2554_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A4: rat,B5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A4),B5)
         => aa_rat_o(aa_rat_fun_rat_o(P,A4),B5) )
     => ( ! [A4: rat,B5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B5),A4)
           => aa_rat_o(aa_rat_fun_rat_o(P,A4),B5) )
       => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2555_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A4: num,B5: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A4),B5)
         => aa_num_o(aa_num_fun_num_o(P,A4),B5) )
     => ( ! [A4: num,B5: num] :
            ( aa_num_o(aa_num_fun_num_o(P,B5),A4)
           => aa_num_o(aa_num_fun_num_o(P,A4),B5) )
       => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2556_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A4),B5)
         => aa_nat_o(aa_nat_fun_nat_o(P,A4),B5) )
     => ( ! [A4: nat,B5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B5),A4)
           => aa_nat_o(aa_nat_fun_nat_o(P,A4),B5) )
       => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2557_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A4: int,B5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A4),B5)
         => aa_int_o(aa_int_fun_int_o(P,A4),B5) )
     => ( ! [A4: int,B5: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B5),A4)
           => aa_int_o(aa_int_fun_int_o(P,A4),B5) )
       => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2558_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2559_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2560_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2561_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2562_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2563_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2564_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2565_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2566_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2567_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2568_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
       => aa_set_nat_o(ord_less_eq_set_nat(C),A) ) ) ).

% dual_order.trans
tff(fact_2569_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% dual_order.trans
tff(fact_2570_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% dual_order.trans
tff(fact_2571_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_2572_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% dual_order.trans
tff(fact_2573_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2574_antisym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2575_antisym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2576_antisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2577_antisym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2578_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2579_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2580_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2581_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2582_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2583_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2584_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2585_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2586_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2587_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_2588_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_2589_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_2590_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X)),aa_int_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_2591_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2592_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2593_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2594_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2595_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2596_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2597_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2598_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2599_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2600_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2601_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2602_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2603_order__eq__refl,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( Xa = Y )
     => aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) ) ).

% order_eq_refl
tff(fact_2604_order__eq__refl,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa = Y )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% order_eq_refl
tff(fact_2605_order__eq__refl,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa = Y )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y) ) ).

% order_eq_refl
tff(fact_2606_order__eq__refl,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ).

% order_eq_refl
tff(fact_2607_order__eq__refl,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ).

% order_eq_refl
tff(fact_2608_linorder__linear,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% linorder_linear
tff(fact_2609_linorder__linear,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% linorder_linear
tff(fact_2610_linorder__linear,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% linorder_linear
tff(fact_2611_linorder__linear,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% linorder_linear
tff(fact_2612_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2613_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2614_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2615_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2616_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2617_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2618_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2619_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2620_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2621_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2622_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2623_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2624_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2625_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2626_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2627_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2628_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2629_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2630_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2631_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2632_linorder__le__cases,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% linorder_le_cases
tff(fact_2633_linorder__le__cases,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% linorder_le_cases
tff(fact_2634_linorder__le__cases,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% linorder_le_cases
tff(fact_2635_linorder__le__cases,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% linorder_le_cases
tff(fact_2636_order__antisym__conv,axiom,
    ! [Y: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),Xa)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_2637_order__antisym__conv,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_2638_order__antisym__conv,axiom,
    ! [Y: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_2639_order__antisym__conv,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_2640_order__antisym__conv,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_2641_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_2642_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_2643_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_2644_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_2645_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% verit_comp_simplify1(2)
tff(fact_2646_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2647_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2648_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2649_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2650_one__reorient,axiom,
    ! [Xa: assn] :
      ( ( one_one_assn = Xa )
    <=> ( Xa = one_one_assn ) ) ).

% one_reorient
tff(fact_2651_one__reorient,axiom,
    ! [Xa: real] :
      ( ( one_one_real = Xa )
    <=> ( Xa = one_one_real ) ) ).

% one_reorient
tff(fact_2652_one__reorient,axiom,
    ! [Xa: rat] :
      ( ( one_one_rat = Xa )
    <=> ( Xa = one_one_rat ) ) ).

% one_reorient
tff(fact_2653_one__reorient,axiom,
    ! [Xa: nat] :
      ( ( one_one_nat = Xa )
    <=> ( Xa = one_one_nat ) ) ).

% one_reorient
tff(fact_2654_one__reorient,axiom,
    ! [Xa: int] :
      ( ( one_one_int = Xa )
    <=> ( Xa = one_one_int ) ) ).

% one_reorient
tff(fact_2655_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_2656_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_2657_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_2658_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_2659_power__increasing,axiom,
    ! [Na: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)) ) ) ).

% power_increasing
tff(fact_2660_power__increasing,axiom,
    ! [Na: nat,N3: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N3)) ) ) ).

% power_increasing
tff(fact_2661_power__increasing,axiom,
    ! [Na: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N3)) ) ) ).

% power_increasing
tff(fact_2662_power__increasing,axiom,
    ! [Na: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)) ) ) ).

% power_increasing
tff(fact_2663_power__increasing,axiom,
    ! [Na: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N3)) ) ) ).

% power_increasing
tff(fact_2664_one__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Na)) ).

% one_le_numeral
tff(fact_2665_one__le__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% one_le_numeral
tff(fact_2666_one__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% one_le_numeral
tff(fact_2667_one__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% one_le_numeral
tff(fact_2668_one__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) ).

% one_le_numeral
tff(fact_2669_one__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% one_le_power
tff(fact_2670_one__le__power,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ).

% one_le_power
tff(fact_2671_one__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% one_le_power
tff(fact_2672_one__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% one_le_power
tff(fact_2673_one__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ).

% one_le_power
tff(fact_2674_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_2675_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_2676_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_2677_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_2678_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_2679_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_2680_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% zle_int
tff(fact_2681_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_2682_nat__mono,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_2683_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_2684_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_2685_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: 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),Z4)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_2686_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(P,X_12)
    <=> ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% all_nat
tff(fact_2687_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
          & aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% ex_nat
tff(fact_2688_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_2689_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_2690_ceiling__mono,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,Xa)) ) ).

% ceiling_mono
tff(fact_2691_ceiling__mono,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,Xa)) ) ).

% ceiling_mono
tff(fact_2692_power__decreasing,axiom,
    ! [Na: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_decreasing
tff(fact_2693_power__decreasing,axiom,
    ! [Na: nat,N3: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N3)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_2694_power__decreasing,axiom,
    ! [Na: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_2695_power__decreasing,axiom,
    ! [Na: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_decreasing
tff(fact_2696_power__decreasing,axiom,
    ! [Na: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_2697_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_2698_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_2699_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_2700_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_2701_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_2702_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_2703_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_2704_pos__mult__pos__ge,axiom,
    ! [Xa: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Na),one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,Na),Xa)) ) ) ).

% pos_mult_pos_ge
tff(fact_2705_le__nat__iff,axiom,
    ! [K: int,Na: 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,Na),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ) ).

% le_nat_iff
tff(fact_2706_nat__leq__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_leq_as_int
tff(fact_2707_inverse__of__nat__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( ( Na != 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,M))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_2708_inverse__of__nat__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( ( Na != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_2709_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2710_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2711_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2712_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2713_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_2714_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_2715_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_2716_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_2717_mult__right__le__one__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Xa),Y)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_2718_mult__right__le__one__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_2719_mult__right__le__one__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Y)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_2720_mult__left__le__one__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Y),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_2721_mult__left__le__one__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_2722_mult__left__le__one__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_2723_discrete,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(A,one_one_Code_integer)),B) ) ).

% discrete
tff(fact_2724_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_2725_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_2726_power__le__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),one_one_real) ) ) ).

% power_le_one
tff(fact_2727_power__le__one,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_2728_power__le__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),one_one_rat) ) ) ).

% power_le_one
tff(fact_2729_power__le__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),one_one_nat) ) ) ).

% power_le_one
tff(fact_2730_power__le__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),one_one_int) ) ) ).

% power_le_one
tff(fact_2731_mod__power__lem,axiom,
    ! [A: int,Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),M)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),zero_zero_int,aa_nat_int(power_power_int(A),Na)) ) ) ).

% mod_power_lem
tff(fact_2732_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_2733_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_2734_nat__le__iff,axiom,
    ! [Xa: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xa)),Na)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_le_iff
tff(fact_2735_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% neg_mod_conj
tff(fact_2736_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_2737_int__mod__eq,axiom,
    ! [B: int,Na: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Na)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Na) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na) )
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Na) = B ) ) ) ) ).

% int_mod_eq
tff(fact_2738_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_2739_int__mod__lem,axiom,
    ! [Na: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),Na) )
      <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na) = B ) ) ) ).

% int_mod_lem
tff(fact_2740_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_2741_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_2742_int__mod__ge,axiom,
    ! [A: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Na)) ) ) ).

% int_mod_ge
tff(fact_2743_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_2744_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_2745_zdiv__mono1,axiom,
    ! [A: int,A3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),B)) ) ) ).

% zdiv_mono1
tff(fact_2746_zdiv__mono2,axiom,
    ! [A: int,B4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B4)) ) ) ) ).

% zdiv_mono2
tff(fact_2747_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_2748_zdiv__mono1__neg,axiom,
    ! [A: int,A3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A3)
     => ( 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,A3),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_2749_zdiv__mono2__neg,axiom,
    ! [A: int,B4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B4)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_2750_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_2751_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_2752_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_2753_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_2754_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_2755_zdiv__le__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),A) ) ) ).

% zdiv_le_dividend
tff(fact_2756_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_2757_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_2758_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_2759_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_2760_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Na) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( Na = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2761_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_2762_nonneg__mod__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% nonneg_mod_div
tff(fact_2763_int__div__less__self,axiom,
    ! [Xa: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa)
     => ( 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,Xa),K)),Xa) ) ) ).

% int_div_less_self
tff(fact_2764_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_2765_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_2766_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_2767_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_2768_zero__le,axiom,
    ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa) ).

% zero_le
tff(fact_2769_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),A3)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_2770_verit__comp__simplify1_I3_J,axiom,
    ! [B4: code_integer,A3: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B4),A3)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A3),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_2771_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B4),A3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_2772_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B4),A3)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_2773_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B4),A3)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_2774_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),A3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_2775_leD,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% leD
tff(fact_2776_leD,axiom,
    ! [Y: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Xa)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y) ) ).

% leD
tff(fact_2777_leD,axiom,
    ! [Y: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),Xa)
     => ~ aa_set_nat_o(ord_less_set_nat(Xa),Y) ) ).

% leD
tff(fact_2778_leD,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ).

% leD
tff(fact_2779_leD,axiom,
    ! [Y: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y) ) ).

% leD
tff(fact_2780_leD,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ).

% leD
tff(fact_2781_leD,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ).

% leD
tff(fact_2782_leI,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ).

% leI
tff(fact_2783_leI,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Xa) ) ).

% leI
tff(fact_2784_leI,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% leI
tff(fact_2785_leI,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% leI
tff(fact_2786_leI,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% leI
tff(fact_2787_leI,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% leI
tff(fact_2788_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2789_nless__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
    <=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2790_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2791_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2792_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2793_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2794_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2795_antisym__conv1,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_2796_antisym__conv1,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_2797_antisym__conv1,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_2798_antisym__conv1,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_2799_antisym__conv1,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_2800_antisym__conv1,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_2801_antisym__conv1,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_2802_antisym__conv2,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_2803_antisym__conv2,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_2804_antisym__conv2,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( ~ aa_set_nat_o(ord_less_set_nat(Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_2805_antisym__conv2,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_2806_antisym__conv2,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_2807_antisym__conv2,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_2808_antisym__conv2,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_2809_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_ge
tff(fact_2810_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_ge
tff(fact_2811_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_le
tff(fact_2812_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_2813_less__le__not__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_2814_less__le__not__le,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_2815_less__le__not__le,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(Y),Xa) ) ) ).

% less_le_not_le
tff(fact_2816_less__le__not__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_2817_less__le__not__le,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_2818_less__le__not__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_2819_less__le__not__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_2820_not__le__imp__less,axiom,
    ! [Y: real,Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% not_le_imp_less
tff(fact_2821_not__le__imp__less,axiom,
    ! [Y: code_integer,Xa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Xa)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y) ) ).

% not_le_imp_less
tff(fact_2822_not__le__imp__less,axiom,
    ! [Y: rat,Xa: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ).

% not_le_imp_less
tff(fact_2823_not__le__imp__less,axiom,
    ! [Y: num,Xa: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y) ) ).

% not_le_imp_less
tff(fact_2824_not__le__imp__less,axiom,
    ! [Y: nat,Xa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ).

% not_le_imp_less
tff(fact_2825_not__le__imp__less,axiom,
    ! [Y: int,Xa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ).

% not_le_imp_less
tff(fact_2826_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2827_order_Oorder__iff__strict,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2828_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_set_nat(A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2829_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2830_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2831_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2832_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2833_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2834_order_Ostrict__iff__order,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2835_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2836_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2837_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2838_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2839_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2840_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans1
tff(fact_2841_order_Ostrict__trans1,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).

% order.strict_trans1
tff(fact_2842_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% order.strict_trans1
tff(fact_2843_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans1
tff(fact_2844_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans1
tff(fact_2845_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_2846_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans1
tff(fact_2847_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans2
tff(fact_2848_order_Ostrict__trans2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).

% order.strict_trans2
tff(fact_2849_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% order.strict_trans2
tff(fact_2850_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans2
tff(fact_2851_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans2
tff(fact_2852_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_2853_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans2
tff(fact_2854_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2855_order_Ostrict__iff__not,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2856_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% order.strict_iff_not
tff(fact_2857_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2858_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2859_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2860_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2861_dense__ge__bounded,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xa)
     => ( ! [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),Xa)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),W2) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_2862_dense__ge__bounded,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xa)
     => ( ! [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),Xa)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_2863_dense__le__bounded,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Y)
             => 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,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_2864_dense__le__bounded,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_2865_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2866_dual__order_Oorder__iff__strict,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2867_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_set_nat(B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2868_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2869_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2870_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2871_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2872_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2873_dual__order_Ostrict__iff__order,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2874_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2875_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2876_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2877_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2878_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2879_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2880_dual__order_Ostrict__trans1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2881_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_set_nat(C),B)
       => aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2882_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2883_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2884_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2885_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2886_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2887_dual__order_Ostrict__trans2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2888_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
       => aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2889_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2890_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2891_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2892_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2893_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2894_dual__order_Ostrict__iff__not,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2895_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2896_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2897_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2898_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2899_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2900_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% order.strict_implies_order
tff(fact_2901_order_Ostrict__implies__order,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% order.strict_implies_order
tff(fact_2902_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% order.strict_implies_order
tff(fact_2903_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% order.strict_implies_order
tff(fact_2904_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_2905_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_2906_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% order.strict_implies_order
tff(fact_2907_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2908_dual__order_Ostrict__implies__order,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2909_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2910_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2911_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2912_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2913_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2914_order__le__less,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_2915_order__le__less,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_2916_order__le__less,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
    <=> ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_2917_order__le__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_2918_order__le__less,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_2919_order__le__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_2920_order__le__less,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_2921_order__less__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_2922_order__less__le,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_2923_order__less__le,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_2924_order__less__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_2925_order__less__le,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_2926_order__less__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_2927_order__less__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_2928_linorder__not__le,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% linorder_not_le
tff(fact_2929_linorder__not__le,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ).

% linorder_not_le
tff(fact_2930_linorder__not__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% linorder_not_le
tff(fact_2931_linorder__not__le,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% linorder_not_le
tff(fact_2932_linorder__not__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% linorder_not_le
tff(fact_2933_linorder__not__le,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% linorder_not_le
tff(fact_2934_linorder__not__less,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ).

% linorder_not_less
tff(fact_2935_linorder__not__less,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Xa) ) ).

% linorder_not_less
tff(fact_2936_linorder__not__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% linorder_not_less
tff(fact_2937_linorder__not__less,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% linorder_not_less
tff(fact_2938_linorder__not__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% linorder_not_less
tff(fact_2939_linorder__not__less,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% linorder_not_less
tff(fact_2940_order__less__imp__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% order_less_imp_le
tff(fact_2941_order__less__imp__le,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y) ) ).

% order_less_imp_le
tff(fact_2942_order__less__imp__le,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
     => aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) ) ).

% order_less_imp_le
tff(fact_2943_order__less__imp__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% order_less_imp_le
tff(fact_2944_order__less__imp__le,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y) ) ).

% order_less_imp_le
tff(fact_2945_order__less__imp__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ).

% order_less_imp_le
tff(fact_2946_order__less__imp__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ).

% order_less_imp_le
tff(fact_2947_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( A != B )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2948_order__le__neq__trans,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( ( A != B )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2949_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( ( A != B )
       => aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).

% order_le_neq_trans
tff(fact_2950_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( A != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2951_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2952_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2953_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2954_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2955_order__neq__le__trans,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != B )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2956_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
       => aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).

% order_neq_le_trans
tff(fact_2957_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2958_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2959_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2960_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2961_order__le__less__trans,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_2962_order__le__less__trans,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_2963_order__le__less__trans,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_set_nat(Y),Z)
       => aa_set_nat_o(ord_less_set_nat(Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_2964_order__le__less__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_2965_order__le__less__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_2966_order__le__less__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_2967_order__le__less__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_2968_order__less__le__trans,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_2969_order__less__le__trans,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_2970_order__less__le__trans,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Y),Z)
       => aa_set_nat_o(ord_less_set_nat(Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_2971_order__less__le__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_2972_order__less__le__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_2973_order__less__le__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_2974_order__less__le__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_2975_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2976_order__le__less__subst1,axiom,
    ! [A: code_integer,F: fun_re6978948838062957830nteger,B: real,C: real] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_real_Code_integer(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,X)),aa_real_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_real_Code_integer(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2977_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2978_order__le__less__subst1,axiom,
    ! [A: code_integer,F: fun_rat_Code_integer,B: rat,C: rat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_rat_Code_integer(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X)),aa_rat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_rat_Code_integer(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2979_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2980_order__le__less__subst1,axiom,
    ! [A: code_integer,F: fun_num_Code_integer,B: num,C: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_num_Code_integer(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(F,X)),aa_num_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_num_Code_integer(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2981_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2982_order__le__less__subst1,axiom,
    ! [A: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X)),aa_nat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_nat_Code_integer(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2983_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2984_order__le__less__subst1,axiom,
    ! [A: code_integer,F: fun_int_Code_integer,B: int,C: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_int_Code_integer(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X)),aa_int_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_int_Code_integer(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_2985_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X)),aa_rat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2986_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_Code_integer,C: code_integer] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_rat_Code_integer(F,X)),aa_rat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2987_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2988_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2989_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2990_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2991_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X)),aa_num_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2992_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_Code_integer,C: code_integer] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(F,X)),aa_num_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2993_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2994_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_2995_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X)),aa_rat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2996_order__less__le__subst1,axiom,
    ! [A: code_integer,F: fun_rat_Code_integer,B: rat,C: rat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_rat_Code_integer(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_rat_Code_integer(F,X)),aa_rat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_rat_Code_integer(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2997_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2998_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_2999_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3000_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3001_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X)),aa_num_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3002_order__less__le__subst1,axiom,
    ! [A: code_integer,F: fun_num_Code_integer,B: num,C: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_num_Code_integer(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(F,X)),aa_num_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_num_Code_integer(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3003_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3004_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y2)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3005_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3006_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_re6978948838062957830nteger,C: code_integer] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_real_Code_integer(F,B)),C)
       => ( ! [X: real,Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,X)),aa_real_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3007_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3008_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_Code_integer,C: code_integer] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_rat_Code_integer(F,B)),C)
       => ( ! [X: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X)),aa_rat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3009_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3010_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_Code_integer,C: code_integer] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(F,B)),C)
       => ( ! [X: num,Y2: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(F,X)),aa_num_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3011_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3012_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,B)),C)
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X)),aa_nat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3013_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C)
       => ( ! [X: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3014_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_Code_integer,C: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,B)),C)
       => ( ! [X: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X)),aa_int_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3015_linorder__le__less__linear,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_3016_linorder__le__less__linear,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
      | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_3017_linorder__le__less__linear,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_3018_linorder__le__less__linear,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_3019_linorder__le__less__linear,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_3020_linorder__le__less__linear,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_3021_order__le__imp__less__or__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3022_order__le__imp__less__or__eq,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3023_order__le__imp__less__or__eq,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3024_order__le__imp__less__or__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3025_order__le__imp__less__or__eq,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3026_order__le__imp__less__or__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3027_order__le__imp__less__or__eq,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3028_le__num__One__iff,axiom,
    ! [Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),one)
    <=> ( Xa = one ) ) ).

% le_num_One_iff
tff(fact_3029_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3030_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3031_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_3032_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_3033_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3034_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3035_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_3036_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_3037_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3038_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3039_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_3040_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_3041_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_3042_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_mono
tff(fact_3043_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_3044_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_3045_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_left_mono
tff(fact_3046_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_left_mono
tff(fact_3047_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_3048_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_3049_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C4: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% less_eqE
tff(fact_3050_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_right_mono
tff(fact_3051_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_right_mono
tff(fact_3052_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_3053_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_3054_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C5: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C5) ) ).

% le_iff_add
tff(fact_3055_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_3056_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_3057_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_3058_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_3059_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_3060_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_3061_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_3062_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_3063_bot_Oextremum,axiom,
    ! [A: filter_nat] : ord_le2510731241096832064er_nat(bot_bot_filter_nat,A) ).

% bot.extremum
tff(fact_3064_bot_Oextremum,axiom,
    ! [A: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A) ).

% bot.extremum
tff(fact_3065_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% bot.extremum
tff(fact_3066_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_3067_bot_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).

% bot.extremum
tff(fact_3068_bot_Oextremum__unique,axiom,
    ! [A: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,bot_bot_filter_nat)
    <=> ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
tff(fact_3069_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_3070_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_3071_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_3072_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_3073_bot_Oextremum__uniqueI,axiom,
    ! [A: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,bot_bot_filter_nat)
     => ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_3074_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_3075_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_3076_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_3077_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_3078_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% less_eq_nat.simps(1)
tff(fact_3079_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_3080_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_3081_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_3082_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_3083_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_3084_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_3085_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_3086_less__numeral__extra_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),one_one_Code_integer) ).

% less_numeral_extra(4)
tff(fact_3087_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_3088_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_imp_le_nat
tff(fact_3089_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_3090_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_or_eq_imp_le
tff(fact_3091_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ( M != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% le_neq_implies_less
tff(fact_3092_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J3)) )
     => ( 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_3093_exists__leI,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ( ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),Na)
           => ~ aa_nat_o(P,N4) )
       => aa_nat_o(P,Na) )
     => ? [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),Na)
          & aa_nat_o(P,N5) ) ) ).

% exists_leI
tff(fact_3094_real__arch__simple,axiom,
    ! [Xa: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_3095_real__arch__simple,axiom,
    ! [Xa: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_3096_inf__sup__ord_I4_J,axiom,
    ! [Y: extended_enat,Xa: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)) ).

% inf_sup_ord(4)
tff(fact_3097_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,Xa: assn] : ord_less_eq_assn(Y,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)) ).

% inf_sup_ord(4)
tff(fact_3098_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,Xa: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)) ).

% inf_sup_ord(4)
tff(fact_3099_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y)) ).

% inf_sup_ord(4)
tff(fact_3100_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)) ).

% inf_sup_ord(4)
tff(fact_3101_inf__sup__ord_I4_J,axiom,
    ! [Y: int,Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)) ).

% inf_sup_ord(4)
tff(fact_3102_inf__sup__ord_I3_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)) ).

% inf_sup_ord(3)
tff(fact_3103_inf__sup__ord_I3_J,axiom,
    ! [Xa: assn,Y: assn] : ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)) ).

% inf_sup_ord(3)
tff(fact_3104_inf__sup__ord_I3_J,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)) ).

% inf_sup_ord(3)
tff(fact_3105_inf__sup__ord_I3_J,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y)) ).

% inf_sup_ord(3)
tff(fact_3106_inf__sup__ord_I3_J,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)) ).

% inf_sup_ord(3)
tff(fact_3107_inf__sup__ord_I3_J,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)) ).

% inf_sup_ord(3)
tff(fact_3108_le__supE,axiom,
    ! [A: extended_enat,B: extended_enat,Xa: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)),Xa)
     => ~ ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),Xa)
         => ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),Xa) ) ) ).

% le_supE
tff(fact_3109_le__supE,axiom,
    ! [A: assn,B: assn,Xa: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B),Xa)
     => ~ ( ord_less_eq_assn(A,Xa)
         => ~ ord_less_eq_assn(B,Xa) ) ) ).

% le_supE
tff(fact_3110_le__supE,axiom,
    ! [A: set_nat,B: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),Xa)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A),Xa)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(B),Xa) ) ) ).

% le_supE
tff(fact_3111_le__supE,axiom,
    ! [A: rat,B: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),Xa)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Xa)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Xa) ) ) ).

% le_supE
tff(fact_3112_le__supE,axiom,
    ! [A: nat,B: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),Xa)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Xa)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Xa) ) ) ).

% le_supE
tff(fact_3113_le__supE,axiom,
    ! [A: int,B: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),Xa)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Xa) ) ) ).

% le_supE
tff(fact_3114_le__supI,axiom,
    ! [A: extended_enat,Xa: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),Xa)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),Xa)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)),Xa) ) ) ).

% le_supI
tff(fact_3115_le__supI,axiom,
    ! [A: assn,Xa: assn,B: assn] :
      ( ord_less_eq_assn(A,Xa)
     => ( ord_less_eq_assn(B,Xa)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B),Xa) ) ) ).

% le_supI
tff(fact_3116_le__supI,axiom,
    ! [A: set_nat,Xa: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),Xa)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),Xa)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),Xa) ) ) ).

% le_supI
tff(fact_3117_le__supI,axiom,
    ! [A: rat,Xa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),Xa) ) ) ).

% le_supI
tff(fact_3118_le__supI,axiom,
    ! [A: nat,Xa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Xa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),Xa) ) ) ).

% le_supI
tff(fact_3119_le__supI,axiom,
    ! [A: int,Xa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Xa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),Xa) ) ) ).

% le_supI
tff(fact_3120_sup__ge1,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)) ).

% sup_ge1
tff(fact_3121_sup__ge1,axiom,
    ! [Xa: assn,Y: assn] : ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)) ).

% sup_ge1
tff(fact_3122_sup__ge1,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)) ).

% sup_ge1
tff(fact_3123_sup__ge1,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y)) ).

% sup_ge1
tff(fact_3124_sup__ge1,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)) ).

% sup_ge1
tff(fact_3125_sup__ge1,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)) ).

% sup_ge1
tff(fact_3126_sup__ge2,axiom,
    ! [Y: extended_enat,Xa: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)) ).

% sup_ge2
tff(fact_3127_sup__ge2,axiom,
    ! [Y: assn,Xa: assn] : ord_less_eq_assn(Y,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)) ).

% sup_ge2
tff(fact_3128_sup__ge2,axiom,
    ! [Y: set_nat,Xa: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)) ).

% sup_ge2
tff(fact_3129_sup__ge2,axiom,
    ! [Y: rat,Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y)) ).

% sup_ge2
tff(fact_3130_sup__ge2,axiom,
    ! [Y: nat,Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)) ).

% sup_ge2
tff(fact_3131_sup__ge2,axiom,
    ! [Y: int,Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)) ).

% sup_ge2
tff(fact_3132_le__supI1,axiom,
    ! [Xa: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),A)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% le_supI1
tff(fact_3133_le__supI1,axiom,
    ! [Xa: assn,A: assn,B: assn] :
      ( ord_less_eq_assn(Xa,A)
     => ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% le_supI1
tff(fact_3134_le__supI1,axiom,
    ! [Xa: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),A)
     => aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI1
tff(fact_3135_le__supI1,axiom,
    ! [Xa: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI1
tff(fact_3136_le__supI1,axiom,
    ! [Xa: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI1
tff(fact_3137_le__supI1,axiom,
    ! [Xa: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI1
tff(fact_3138_le__supI2,axiom,
    ! [Xa: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),B)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% le_supI2
tff(fact_3139_le__supI2,axiom,
    ! [Xa: assn,B: assn,A: assn] :
      ( ord_less_eq_assn(Xa,B)
     => ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% le_supI2
tff(fact_3140_le__supI2,axiom,
    ! [Xa: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),B)
     => aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI2
tff(fact_3141_le__supI2,axiom,
    ! [Xa: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI2
tff(fact_3142_le__supI2,axiom,
    ! [Xa: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI2
tff(fact_3143_le__supI2,axiom,
    ! [Xa: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI2
tff(fact_3144_sup_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(D),B)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,C),D)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ) ).

% sup.mono
tff(fact_3145_sup_Omono,axiom,
    ! [C: assn,A: assn,D: assn,B: assn] :
      ( ord_less_eq_assn(C,A)
     => ( ord_less_eq_assn(D,B)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).

% sup.mono
tff(fact_3146_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D),B)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% sup.mono
tff(fact_3147_sup_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ) ).

% sup.mono
tff(fact_3148_sup_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% sup.mono
tff(fact_3149_sup_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% sup.mono
tff(fact_3150_sup__mono,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat,D: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),C)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),D)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,C),D)) ) ) ).

% sup_mono
tff(fact_3151_sup__mono,axiom,
    ! [A: assn,C: assn,B: assn,D: assn] :
      ( ord_less_eq_assn(A,C)
     => ( ord_less_eq_assn(B,D)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D)) ) ) ).

% sup_mono
tff(fact_3152_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),C)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),D)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D)) ) ) ).

% sup_mono
tff(fact_3153_sup__mono,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D)) ) ) ).

% sup_mono
tff(fact_3154_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D)) ) ) ).

% sup_mono
tff(fact_3155_sup__mono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D)) ) ) ).

% sup_mono
tff(fact_3156_sup__least,axiom,
    ! [Y: extended_enat,Xa: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y),Xa)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Xa)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Z)),Xa) ) ) ).

% sup_least
tff(fact_3157_sup__least,axiom,
    ! [Y: assn,Xa: assn,Z: assn] :
      ( ord_less_eq_assn(Y,Xa)
     => ( ord_less_eq_assn(Z,Xa)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z),Xa) ) ) ).

% sup_least
tff(fact_3158_sup__least,axiom,
    ! [Y: set_nat,Xa: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),Xa)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Z),Xa)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),Xa) ) ) ).

% sup_least
tff(fact_3159_sup__least,axiom,
    ! [Y: rat,Xa: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z)),Xa) ) ) ).

% sup_least
tff(fact_3160_sup__least,axiom,
    ! [Y: nat,Xa: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),Xa) ) ) ).

% sup_least
tff(fact_3161_sup__least,axiom,
    ! [Y: int,Xa: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),Xa) ) ) ).

% sup_least
tff(fact_3162_le__iff__sup,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),Y)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3163_le__iff__sup,axiom,
    ! [Xa: assn,Y: assn] :
      ( ord_less_eq_assn(Xa,Y)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3164_le__iff__sup,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3165_le__iff__sup,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3166_le__iff__sup,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3167_le__iff__sup,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3168_sup_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
     => ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) ) ) ).

% sup.orderE
tff(fact_3169_sup_OorderE,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
     => ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).

% sup.orderE
tff(fact_3170_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.orderE
tff(fact_3171_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.orderE
tff(fact_3172_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.orderE
tff(fact_3173_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.orderE
tff(fact_3174_sup_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) )
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A) ) ).

% sup.orderI
tff(fact_3175_sup_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
     => ord_less_eq_assn(B,A) ) ).

% sup.orderI
tff(fact_3176_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
     => aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ).

% sup.orderI
tff(fact_3177_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% sup.orderI
tff(fact_3178_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% sup.orderI
tff(fact_3179_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% sup.orderI
tff(fact_3180_sup__unique,axiom,
    ! [F: fun_Ex8341929632020906428d_enat,Xa: extended_enat,Y: extended_enat] :
      ( ! [X: extended_enat,Y2: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(X),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(F,X),Y2))
     => ( ! [X: extended_enat,Y2: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y2),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(F,X),Y2))
       => ( ! [X: extended_enat,Y2: extended_enat,Z3: extended_enat] :
              ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y2),X)
             => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z3),X)
               => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(F,Y2),Z3)),X) ) )
         => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(F,Xa),Y) ) ) ) ) ).

% sup_unique
tff(fact_3181_sup__unique,axiom,
    ! [F: fun_as3983628580203722482n_assn,Xa: assn,Y: assn] :
      ( ! [X: assn,Y2: assn] : ord_less_eq_assn(X,aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y2))
     => ( ! [X: assn,Y2: assn] : ord_less_eq_assn(Y2,aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y2))
       => ( ! [X: assn,Y2: assn,Z3: assn] :
              ( ord_less_eq_assn(Y2,X)
             => ( ord_less_eq_assn(Z3,X)
               => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(F,Y2),Z3),X) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,Xa),Y) ) ) ) ) ).

% sup_unique
tff(fact_3182_sup__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,Xa: set_nat,Y: set_nat] :
      ( ! [X: set_nat,Y2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y2))
     => ( ! [X: set_nat,Y2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Y2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y2))
       => ( ! [X: set_nat,Y2: set_nat,Z3: set_nat] :
              ( aa_set_nat_o(ord_less_eq_set_nat(Y2),X)
             => ( aa_set_nat_o(ord_less_eq_set_nat(Z3),X)
               => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z3)),X) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Xa),Y) ) ) ) ) ).

% sup_unique
tff(fact_3183_sup__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,Xa: rat,Y: rat] :
      ( ! [X: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y2))
     => ( ! [X: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y2))
       => ( ! [X: rat,Y2: rat,Z3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),X)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,Y2),Z3)),X) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,Xa),Y) ) ) ) ) ).

% sup_unique
tff(fact_3184_sup__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,Xa: nat,Y: nat] :
      ( ! [X: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y2))
     => ( ! [X: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y2))
       => ( ! [X: nat,Y2: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),X)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z3)),X) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,Xa),Y) ) ) ) ) ).

% sup_unique
tff(fact_3185_sup__unique,axiom,
    ! [F: fun_int_fun_int_int,Xa: int,Y: int] :
      ( ! [X: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(F,X),Y2))
     => ( ! [X: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),aa_int_int(aa_int_fun_int_int(F,X),Y2))
       => ( ! [X: int,Y2: int,Z3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),X)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y2),Z3)),X) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y) = aa_int_int(aa_int_fun_int_int(F,Xa),Y) ) ) ) ) ).

% sup_unique
tff(fact_3186_sup_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3187_sup_Oabsorb1,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3188_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3189_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3190_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3191_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_3192_sup_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3193_sup_Oabsorb2,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3194_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3195_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3196_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3197_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_3198_sup__absorb1,axiom,
    ! [Y: extended_enat,Xa: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y),Xa)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) = Xa ) ) ).

% sup_absorb1
tff(fact_3199_sup__absorb1,axiom,
    ! [Y: assn,Xa: assn] :
      ( ord_less_eq_assn(Y,Xa)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) = Xa ) ) ).

% sup_absorb1
tff(fact_3200_sup__absorb1,axiom,
    ! [Y: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),Xa)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) = Xa ) ) ).

% sup_absorb1
tff(fact_3201_sup__absorb1,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y) = Xa ) ) ).

% sup_absorb1
tff(fact_3202_sup__absorb1,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y) = Xa ) ) ).

% sup_absorb1
tff(fact_3203_sup__absorb1,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y) = Xa ) ) ).

% sup_absorb1
tff(fact_3204_sup__absorb2,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),Y)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3205_sup__absorb2,axiom,
    ! [Xa: assn,Y: assn] :
      ( ord_less_eq_assn(Xa,Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3206_sup__absorb2,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3207_sup__absorb2,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3208_sup__absorb2,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3209_sup__absorb2,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3210_sup_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,B),C)),A)
     => ~ ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
         => ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A) ) ) ).

% sup.boundedE
tff(fact_3211_sup_OboundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C),A)
     => ~ ( ord_less_eq_assn(B,A)
         => ~ ord_less_eq_assn(C,A) ) ) ).

% sup.boundedE
tff(fact_3212_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(C),A) ) ) ).

% sup.boundedE
tff(fact_3213_sup_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% sup.boundedE
tff(fact_3214_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% sup.boundedE
tff(fact_3215_sup_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% sup.boundedE
tff(fact_3216_sup_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3217_sup_OboundedI,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ord_less_eq_assn(B,A)
     => ( ord_less_eq_assn(C,A)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C),A) ) ) ).

% sup.boundedI
tff(fact_3218_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3219_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3220_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3221_sup_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_3222_sup_Oorder__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
    <=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) ) ) ).

% sup.order_iff
tff(fact_3223_sup_Oorder__iff,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).

% sup.order_iff
tff(fact_3224_sup_Oorder__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_3225_sup_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.order_iff
tff(fact_3226_sup_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_3227_sup_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.order_iff
tff(fact_3228_sup_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ).

% sup.cobounded1
tff(fact_3229_sup_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ord_less_eq_assn(A,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).

% sup.cobounded1
tff(fact_3230_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded1
tff(fact_3231_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ).

% sup.cobounded1
tff(fact_3232_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded1
tff(fact_3233_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded1
tff(fact_3234_sup_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ).

% sup.cobounded2
tff(fact_3235_sup_Ocobounded2,axiom,
    ! [B: assn,A: assn] : ord_less_eq_assn(B,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).

% sup.cobounded2
tff(fact_3236_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded2
tff(fact_3237_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ).

% sup.cobounded2
tff(fact_3238_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded2
tff(fact_3239_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded2
tff(fact_3240_sup_Oabsorb__iff1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3241_sup_Oabsorb__iff1,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3242_sup_Oabsorb__iff1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3243_sup_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3244_sup_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3245_sup_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_3246_sup_Oabsorb__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3247_sup_Oabsorb__iff2,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3248_sup_Oabsorb__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3249_sup_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3250_sup_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3251_sup_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_3252_sup_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3253_sup_OcoboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( ord_less_eq_assn(C,A)
     => ord_less_eq_assn(C,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.coboundedI1
tff(fact_3254_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
     => aa_set_nat_o(ord_less_eq_set_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3255_sup_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3256_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_3257_sup_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI1
tff(fact_3258_sup_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),B)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3259_sup_OcoboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( ord_less_eq_assn(C,B)
     => ord_less_eq_assn(C,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.coboundedI2
tff(fact_3260_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
     => aa_set_nat_o(ord_less_eq_set_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3261_sup_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3262_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_3263_sup_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI2
tff(fact_3264_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3265_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3266_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3267_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3268_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3269_mult_Ocomm__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% mult.comm_neutral
tff(fact_3270_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_3271_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_3272_mult_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.comm_neutral
tff(fact_3273_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_3274_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% add_leE
tff(fact_3275_le__add1,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) ).

% le_add1
tff(fact_3276_le__add2,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% le_add2
tff(fact_3277_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% add_leD1
tff(fact_3278_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% add_leD2
tff(fact_3279_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_3280_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_3281_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_3282_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_le_add1
tff(fact_3283_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_le_add2
tff(fact_3284_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ? [K3: nat] : Na = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_3285_insert__mono,axiom,
    ! [C2: set_int,D3: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(C2),D3)
     => aa_set_int_o(ord_less_eq_set_int(insert_int(A,C2)),insert_int(A,D3)) ) ).

% insert_mono
tff(fact_3286_insert__mono,axiom,
    ! [C2: set_VEBT_VEBT,D3: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(C2),D3)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(A,C2)),insert_VEBT_VEBT(A,D3)) ) ).

% insert_mono
tff(fact_3287_insert__mono,axiom,
    ! [C2: set_real,D3: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(C2),D3)
     => aa_set_real_o(ord_less_eq_set_real(insert_real(A,C2)),insert_real(A,D3)) ) ).

% insert_mono
tff(fact_3288_insert__mono,axiom,
    ! [C2: set_nat,D3: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C2),D3)
     => aa_set_nat_o(ord_less_eq_set_nat(insert_nat(A,C2)),insert_nat(A,D3)) ) ).

% insert_mono
tff(fact_3289_subset__insert,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,B2))
      <=> aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ) ).

% subset_insert
tff(fact_3290_subset__insert,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,B2))
      <=> aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ) ).

% subset_insert
tff(fact_3291_subset__insert,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(Xa,B2))
      <=> aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2) ) ) ).

% subset_insert
tff(fact_3292_subset__insert,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xa,B2))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ) ).

% subset_insert
tff(fact_3293_subset__insertI,axiom,
    ! [B2: set_int,A: int] : aa_set_int_o(ord_less_eq_set_int(B2),insert_int(A,B2)) ).

% subset_insertI
tff(fact_3294_subset__insertI,axiom,
    ! [B2: set_VEBT_VEBT,A: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B2),insert_VEBT_VEBT(A,B2)) ).

% subset_insertI
tff(fact_3295_subset__insertI,axiom,
    ! [B2: set_real,A: real] : aa_set_real_o(ord_less_eq_set_real(B2),insert_real(A,B2)) ).

% subset_insertI
tff(fact_3296_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : aa_set_nat_o(ord_less_eq_set_nat(B2),insert_nat(A,B2)) ).

% subset_insertI
tff(fact_3297_subset__insertI2,axiom,
    ! [A2: set_int,B2: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,B2)) ) ).

% subset_insertI2
tff(fact_3298_subset__insertI2,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(B,B2)) ) ).

% subset_insertI2
tff(fact_3299_subset__insertI2,axiom,
    ! [A2: set_real,B2: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,B2)) ) ).

% subset_insertI2
tff(fact_3300_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(B,B2)) ) ).

% subset_insertI2
tff(fact_3301_less__eq__real__def,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_real_def
tff(fact_3302_div__le__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),M) ).

% div_le_dividend
tff(fact_3303_div__le__mono,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),K)) ) ).

% div_le_mono
tff(fact_3304_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_cube
tff(fact_3305_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_3306_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_3307_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_3308_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_3309_Un__mono,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat,D3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),D3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D3)) ) ) ).

% Un_mono
tff(fact_3310_Un__least,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C2) ) ) ).

% Un_least
tff(fact_3311_Un__upper1,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% Un_upper1
tff(fact_3312_Un__upper2,axiom,
    ! [B2: set_nat,A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% Un_upper2
tff(fact_3313_Un__absorb1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_3314_Un__absorb2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = A2 ) ) ).

% Un_absorb2
tff(fact_3315_subset__UnE,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
     => ~ ! [A5: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(A5),A2)
           => ! [B6: set_nat] :
                ( aa_set_nat_o(ord_less_eq_set_nat(B6),B2)
               => ( C2 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A5),B6) ) ) ) ) ).

% subset_UnE
tff(fact_3316_subset__Un__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_3317_mod__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),M) ).

% mod_less_eq_dividend
tff(fact_3318_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
         => aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% psubsetE
tff(fact_3319_psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_3320_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% psubset_imp_subset
tff(fact_3321_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
       => aa_set_nat_o(ord_less_set_nat(A2),C2) ) ) ).

% psubset_subset_trans
tff(fact_3322_subset__not__subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% subset_not_subset_eq
tff(fact_3323_subset__psubset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_set_nat(B2),C2)
       => aa_set_nat_o(ord_less_set_nat(A2),C2) ) ) ).

% subset_psubset_trans
tff(fact_3324_subset__iff__psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_set_nat(A2),B2)
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_3325_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_3326_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_3327_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_3328_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_3329_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_3330_unset__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),Na) = aa_int_nat(nat2,bit_se4203085406695923979it_int(M,aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% unset_bit_nat_def
tff(fact_3331_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),Na) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_3332_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_3333_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),times_3573771949741848930nteger(C,B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3334_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3335_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_3336_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3337_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(C,A)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3338_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3339_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3340_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_3341_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),times_3573771949741848930nteger(B,C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3342_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3343_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_3344_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3345_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(A,C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3346_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3347_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3348_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_3349_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),times_3573771949741848930nteger(C,B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3350_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3351_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_3352_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3353_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(C,A)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3354_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3355_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3356_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_3357_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),times_3573771949741848930nteger(B,C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3358_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3359_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_3360_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3361_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3362_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3363_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3364_field__le__mult__one__interval,axiom,
    ! [Xa: real,Y: real] :
      ( ! [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z3),Xa)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% field_le_mult_one_interval
tff(fact_3365_field__le__mult__one__interval,axiom,
    ! [Xa: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),Xa)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% field_le_mult_one_interval
tff(fact_3366_convex__bound__le,axiom,
    ! [Xa: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(U),Xa)),aa_real_real(times_times_real(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3367_convex__bound__le,axiom,
    ! [Xa: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3368_convex__bound__le,axiom,
    ! [Xa: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3369_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3370_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3371_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3372_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3373_self__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% self_le_power
tff(fact_3374_self__le__power,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ) ).

% self_le_power
tff(fact_3375_self__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% self_le_power
tff(fact_3376_self__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% self_le_power
tff(fact_3377_self__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Na)) ) ) ).

% self_le_power
tff(fact_3378_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Na),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_3379_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(times_7803423173614009249d_enat,M),Na))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_3380_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_3381_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_3382_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_3383_nat__add__distrib,axiom,
    ! [Z: int,Z4: 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),Z4)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_3384_lambda__one,axiom,
    aTP_Lamm_cd = aa_ass954072293579886259n_assn(times_times_assn,one_one_assn) ).

% lambda_one
tff(fact_3385_lambda__one,axiom,
    aTP_Lamm_ce = times_times_real(one_one_real) ).

% lambda_one
tff(fact_3386_lambda__one,axiom,
    aTP_Lamm_cf = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).

% lambda_one
tff(fact_3387_lambda__one,axiom,
    aTP_Lamm_cg = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_3388_lambda__one,axiom,
    aTP_Lamm_ch = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_3389_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q4: int,R3: int,Q2: int,R: 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),Q4)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q4) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3390_unique__quotient__lemma,axiom,
    ! [B: int,Q4: int,R3: int,Q2: int,R: 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),Q4)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3391_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R: int,B4: int,Q4: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R3)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3392_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R: int,B4: int,Q4: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R3))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B4)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3393_q__pos__lemma,axiom,
    ! [B4: int,Q4: int,R3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R3))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q4) ) ) ) ).

% q_pos_lemma
tff(fact_3394_nat__mult__distrib,axiom,
    ! [Z: int,Z4: 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),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_3395_int__mod__ge_H,axiom,
    ! [B: int,Na: 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),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Na)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na)) ) ) ).

% int_mod_ge'
tff(fact_3396_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_3397_nat__div__distrib_H,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_3398_nat__div__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_3399_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_3400_nat__mod__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_3401_convex__bound__lt,axiom,
    ! [Xa: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(U),Xa)),aa_real_real(times_times_real(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3402_convex__bound__lt,axiom,
    ! [Xa: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V)
           => ( ( plus_p5714425477246183910nteger(U,V) = one_one_Code_integer )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(times_3573771949741848930nteger(U,Xa),times_3573771949741848930nteger(V,Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3403_convex__bound__lt,axiom,
    ! [Xa: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3404_convex__bound__lt,axiom,
    ! [Xa: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3405_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_3406_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3407_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3408_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(times_times_real(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B))) ) ) ).

% mult_ceiling_le
tff(fact_3409_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))) ) ) ).

% mult_ceiling_le
tff(fact_3410_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Xa),Z)),aa_real_real(times_times_real(Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3411_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(Xa,Z)),times_3573771949741848930nteger(Y,Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3412_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3413_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3414_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),Xa)),aa_real_real(times_times_real(Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3415_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(Z,Xa)),times_3573771949741848930nteger(Z,Y))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3416_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3417_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3418_pos__mod__sign2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% pos_mod_sign2
tff(fact_3419_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_3420_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_3421_nmod2,axiom,
    ! [Na: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),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,Na),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% nmod2
tff(fact_3422_zero__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Na)) ).

% zero_le_numeral
tff(fact_3423_zero__le__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% zero_le_numeral
tff(fact_3424_zero__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_le_numeral
tff(fact_3425_zero__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_le_numeral
tff(fact_3426_zero__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_le_numeral
tff(fact_3427_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_3428_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),zero_z3403309356797280102nteger) ).

% not_numeral_le_zero
tff(fact_3429_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_3430_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_3431_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_3432_real__archimedian__rdiv__eq__0,axiom,
    ! [Xa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M4)),Xa)),C) )
         => ( Xa = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_3433_add__nonpos__eq__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y) = zero_zero_real )
        <=> ( ( Xa = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3434_add__nonpos__eq__0__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3435_add__nonpos__eq__0__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3436_add__nonpos__eq__0__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3437_add__nonneg__eq__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y) = zero_zero_real )
        <=> ( ( Xa = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3438_add__nonneg__eq__0__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3439_add__nonneg__eq__0__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3440_add__nonneg__eq__0__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3441_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_3442_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_3443_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_3444_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_3445_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3446_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3447_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3448_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3449_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing2
tff(fact_3450_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing2
tff(fact_3451_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_3452_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_3453_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3454_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3455_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3456_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3457_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing
tff(fact_3458_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing
tff(fact_3459_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_3460_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_3461_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3462_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3463_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3464_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3465_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3466_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3467_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3468_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3469_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3470_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3471_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3472_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3473_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),A)) ).

% zero_le_square
tff(fact_3474_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)) ).

% zero_le_square
tff(fact_3475_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_3476_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ).

% split_mult_pos_le
tff(fact_3477_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).

% split_mult_pos_le
tff(fact_3478_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% split_mult_pos_le
tff(fact_3479_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3480_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3481_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3482_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3483_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3484_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3485_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono
tff(fact_3486_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono
tff(fact_3487_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_3488_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_3489_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3490_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3491_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3492_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono
tff(fact_3493_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono
tff(fact_3494_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_3495_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_3496_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3497_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3498_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3499_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_3500_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_3501_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_3502_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_3503_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3504_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3505_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3506_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3507_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_3508_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3509_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3510_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_3511_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_3512_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3513_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3514_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_3515_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3516_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3517_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3518_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3519_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_3520_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_3521_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_3522_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3523_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3524_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3525_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3526_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3527_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I),J)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J,L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3528_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3529_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_3530_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_3531_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3532_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),J)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J,L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3533_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3534_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_3535_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_3536_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_3537_add__le__less__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,D)) ) ) ).

% add_le_less_mono
tff(fact_3538_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_3539_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_3540_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_3541_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_3542_add__less__le__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,D)) ) ) ).

% add_less_le_mono
tff(fact_3543_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_3544_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_3545_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_3546_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_3547_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_3548_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_right_mono
tff(fact_3549_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) ) ) ).

% divide_right_mono
tff(fact_3550_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_3551_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_3552_divide__nonneg__nonneg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xa),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3553_divide__nonneg__nonneg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3554_divide__nonneg__nonpos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xa),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_3555_divide__nonneg__nonpos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_3556_divide__nonpos__nonneg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xa),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_3557_divide__nonpos__nonneg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_3558_divide__nonpos__nonpos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xa),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3559_divide__nonpos__nonpos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3560_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(divide_divide_real(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_3561_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_3562_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_3563_zero__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_le_power
tff(fact_3564_zero__le__power,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ).

% zero_le_power
tff(fact_3565_zero__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_le_power
tff(fact_3566_zero__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_le_power
tff(fact_3567_zero__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_le_power
tff(fact_3568_power__mono,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ).

% power_mono
tff(fact_3569_power__mono,axiom,
    ! [A: code_integer,B: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na)) ) ) ).

% power_mono
tff(fact_3570_power__mono,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono
tff(fact_3571_power__mono,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ).

% power_mono
tff(fact_3572_power__mono,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono
tff(fact_3573_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_0_le_iff
tff(fact_3574_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_0_le_iff
tff(fact_3575_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_0_le_iff
tff(fact_3576_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_0_le_iff
tff(fact_3577_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_0_le_iff
tff(fact_3578_split__zmod,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Na) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J2),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J2) ) )
             => aa_int_o(P,J2) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J2) ) )
             => aa_int_o(P,J2) ) ) ) ) ).

% split_zmod
tff(fact_3579_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3580_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3581_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3582_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3583_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3584_split__zdiv,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J2),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J2) ) )
             => aa_int_o(P,I2) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J2) ) )
             => aa_int_o(P,I2) ) ) ) ) ).

% split_zdiv
tff(fact_3585_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3586_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3587_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_3588_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_3589_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_3590_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_3591_zero__less__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one
tff(fact_3592_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_3593_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_3594_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_3595_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_3596_not__one__less__zero,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),zero_z3403309356797280102nteger) ).

% not_one_less_zero
tff(fact_3597_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_3598_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_3599_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_3600_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_3601_less__numeral__extra_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% less_numeral_extra(1)
tff(fact_3602_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3603_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),one_one_real) ).

% not_numeral_less_one
tff(fact_3604_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat) ).

% not_numeral_less_one
tff(fact_3605_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) ).

% not_numeral_less_one
tff(fact_3606_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) ).

% not_numeral_less_one
tff(fact_3607_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),one_one_Code_integer) ).

% not_numeral_less_one
tff(fact_3608_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_3609_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_3610_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_3611_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_3612_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_3613_numeral__One,axiom,
    aa_num_Code_integer(numera6620942414471956472nteger,one) = one_one_Code_integer ).

% numeral_One
tff(fact_3614_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_3615_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)) ).

% less_add_one
tff(fact_3616_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_3617_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_3618_less__add__one,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),plus_p5714425477246183910nteger(A,one_one_Code_integer)) ).

% less_add_one
tff(fact_3619_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real)) ) ).

% add_mono1
tff(fact_3620_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat)) ) ).

% add_mono1
tff(fact_3621_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_3622_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_3623_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,one_one_Code_integer)),plus_p5714425477246183910nteger(B,one_one_Code_integer)) ) ).

% add_mono1
tff(fact_3624_one__plus__numeral__commute,axiom,
    ! [Xa: num] : plus_plus_complex(one_one_complex,numera6690914467698888265omplex(Xa)) = plus_plus_complex(numera6690914467698888265omplex(Xa),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_3625_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xa)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_3626_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Xa)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Xa)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_3627_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xa)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_3628_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xa)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_3629_one__plus__numeral__commute,axiom,
    ! [Xa: num] : plus_p5714425477246183910nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Xa)) = plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa),one_one_Code_integer) ).

% one_plus_numeral_commute
tff(fact_3630_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_3631_less__1__mult,axiom,
    ! [M: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(M),Na)) ) ) ).

% less_1_mult
tff(fact_3632_less__1__mult,axiom,
    ! [M: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),Na)) ) ) ).

% less_1_mult
tff(fact_3633_less__1__mult,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% less_1_mult
tff(fact_3634_less__1__mult,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),Na)) ) ) ).

% less_1_mult
tff(fact_3635_less__1__mult,axiom,
    ! [M: code_integer,Na: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Na)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),times_3573771949741848930nteger(M,Na)) ) ) ).

% less_1_mult
tff(fact_3636_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex(A,B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_3637_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_3638_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_3639_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M4)),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,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))) ) ).

% mono_nat_linear_lb
tff(fact_3640_subset__singletonD,axiom,
    ! [A2: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT))
     => ( ( A2 = bot_bo8194388402131092736T_VEBT )
        | ( A2 = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singletonD
tff(fact_3641_subset__singletonD,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,bot_bot_set_real))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = insert_real(Xa,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_3642_subset__singletonD,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,bot_bot_set_int))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int(Xa,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_3643_subset__singletonD,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xa,bot_bot_set_nat))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat(Xa,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_3644_subset__singleton__iff,axiom,
    ! [X7: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(X7),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))
    <=> ( ( X7 = bot_bo8194388402131092736T_VEBT )
        | ( X7 = insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singleton_iff
tff(fact_3645_subset__singleton__iff,axiom,
    ! [X7: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(X7),insert_real(A,bot_bot_set_real))
    <=> ( ( X7 = bot_bot_set_real )
        | ( X7 = insert_real(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_3646_subset__singleton__iff,axiom,
    ! [X7: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X7),insert_int(A,bot_bot_set_int))
    <=> ( ( X7 = bot_bot_set_int )
        | ( X7 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_3647_subset__singleton__iff,axiom,
    ! [X7: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X7),insert_nat(A,bot_bot_set_nat))
    <=> ( ( X7 = bot_bot_set_nat )
        | ( X7 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_3648_left__right__inverse__power,axiom,
    ! [Xa: assn,Y: assn,Na: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Xa),Y) = one_one_assn )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(Xa),Na)),aa_nat_assn(power_power_assn(Y),Na)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_3649_left__right__inverse__power,axiom,
    ! [Xa: complex,Y: complex,Na: nat] :
      ( ( aa_complex_complex(times_times_complex(Xa),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xa),Na)),aa_nat_complex(power_power_complex(Y),Na)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_3650_left__right__inverse__power,axiom,
    ! [Xa: code_integer,Y: code_integer,Na: nat] :
      ( ( times_3573771949741848930nteger(Xa,Y) = one_one_Code_integer )
     => ( times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Na),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Na)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_3651_left__right__inverse__power,axiom,
    ! [Xa: real,Y: real,Na: nat] :
      ( ( aa_real_real(times_times_real(Xa),Y) = one_one_real )
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_3652_left__right__inverse__power,axiom,
    ! [Xa: rat,Y: rat,Na: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xa),Na)),aa_nat_rat(power_power_rat(Y),Na)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_3653_left__right__inverse__power,axiom,
    ! [Xa: nat,Y: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Na)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_3654_left__right__inverse__power,axiom,
    ! [Xa: int,Y: int,Na: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xa),Na)),aa_nat_int(power_power_int(Y),Na)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_3655_power__one__over,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(divide1717551699836669952omplex(one_one_complex,A)),Na) = divide1717551699836669952omplex(one_one_complex,aa_nat_complex(power_power_complex(A),Na)) ).

% power_one_over
tff(fact_3656_power__one__over,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),A)),Na) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ).

% power_one_over
tff(fact_3657_power__one__over,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),Na) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ).

% power_one_over
tff(fact_3658_power__0,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),zero_zero_nat) = one_one_assn ).

% power_0
tff(fact_3659_power__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_3660_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_3661_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_3662_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_3663_power__0,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_3664_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_3665_mlex__leI,axiom,
    ! [A: nat,A3: nat,B: nat,B4: nat,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N3)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N3)),B4)) ) ) ).

% mlex_leI
tff(fact_3666_div__mult__le,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),A) ).

% div_mult_le
tff(fact_3667_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))),M) ).

% times_div_less_eq_dividend
tff(fact_3668_div__times__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na)),M) ).

% div_times_less_eq_dividend
tff(fact_3669_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z)
    <=> ? [N2: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% zle_iff_zadd
tff(fact_3670_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_3671_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_3672_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_3673_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_3674_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_3675_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_3676_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_3677_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
    <=> ? [Q5: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q5) ) ).

% zmod_eq_0_iff
tff(fact_3678_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
     => ? [Q3: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_3679_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_3680_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_3681_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_3682_axxmod2,axiom,
    ! [Xa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Xa)),Xa)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int )
      & ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Xa)),Xa)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% axxmod2
tff(fact_3683_axxdiv2,axiom,
    ! [Xa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Xa)),Xa)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Xa )
      & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Xa)),Xa)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Xa ) ) ).

% axxdiv2
tff(fact_3684_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: 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,Na),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
      <=> ! [I2: int,J2: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J2),K)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J2) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) ) ) ).

% split_pos_lemma
tff(fact_3685_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: 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,Na),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
      <=> ! [I2: int,J2: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),zero_zero_int)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J2) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) ) ) ).

% split_neg_lemma
tff(fact_3686_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_3687_add__neg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_nonpos
tff(fact_3688_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_3689_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_3690_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_3691_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3692_add__nonneg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,B)) ) ) ).

% add_nonneg_pos
tff(fact_3693_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3694_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3695_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3696_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_3697_add__nonpos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_neg
tff(fact_3698_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_3699_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_3700_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_3701_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3702_add__pos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,B)) ) ) ).

% add_pos_nonneg
tff(fact_3703_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3704_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3705_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3706_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3707_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),plus_p5714425477246183910nteger(A,C)) ) ) ).

% add_strict_increasing
tff(fact_3708_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3709_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3710_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3711_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3712_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),plus_p5714425477246183910nteger(A,C)) ) ) ).

% add_strict_increasing2
tff(fact_3713_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3714_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3715_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3716_field__le__epsilon,axiom,
    ! [Xa: real,Y: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% field_le_epsilon
tff(fact_3717_field__le__epsilon,axiom,
    ! [Xa: rat,Y: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% field_le_epsilon
tff(fact_3718_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),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,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3719_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3720_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3721_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3722_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),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,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3723_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3724_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3725_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3726_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),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,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3727_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3728_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3729_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3730_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3731_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3732_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3733_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3734_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3735_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3736_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),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,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3737_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3738_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3739_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3740_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),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,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3741_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3742_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3743_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3744_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3745_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3746_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3747_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3748_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3749_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3750_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),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,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3751_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3752_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3753_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3754_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3755_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3756_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3757_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3758_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3759_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3760_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3761_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3762_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),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,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3763_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(C,A)),times_3573771949741848930nteger(C,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3764_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3765_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3766_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3767_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),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,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3768_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3769_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3770_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3771_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3772_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3773_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3774_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3775_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3776_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3777_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3778_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,C)),times_3573771949741848930nteger(B,D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3779_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3780_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3781_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3782_sum__squares__ge__zero,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xa),Xa)),aa_real_real(times_times_real(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3783_sum__squares__ge__zero,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3784_sum__squares__ge__zero,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3785_sum__squares__le__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xa),Xa)),aa_real_real(times_times_real(Y),Y))),zero_zero_real)
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3786_sum__squares__le__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3787_sum__squares__le__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3788_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3789_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3790_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3791_div__positive,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),divide6298287555418463151nteger(A,B)) ) ) ).

% div_positive
tff(fact_3792_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_3793_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_positive
tff(fact_3794_frac__le,axiom,
    ! [Y: real,Xa: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xa),Z)),aa_real_real(divide_divide_real(Y),W)) ) ) ) ) ).

% frac_le
tff(fact_3795_frac__le,axiom,
    ! [Y: rat,Xa: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W)) ) ) ) ) ).

% frac_le
tff(fact_3796_frac__less,axiom,
    ! [Xa: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xa),Z)),aa_real_real(divide_divide_real(Y),W)) ) ) ) ) ).

% frac_less
tff(fact_3797_frac__less,axiom,
    ! [Xa: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W)) ) ) ) ) ).

% frac_less
tff(fact_3798_frac__less2,axiom,
    ! [Xa: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xa),Z)),aa_real_real(divide_divide_real(Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_3799_frac__less2,axiom,
    ! [Xa: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_3800_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),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,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3801_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_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,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3802_divide__nonneg__neg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xa),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_3803_divide__nonneg__neg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3804_divide__nonneg__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xa),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3805_divide__nonneg__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3806_divide__nonpos__neg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xa),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3807_divide__nonpos__neg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3808_divide__nonpos__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xa),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_3809_divide__nonpos__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3810_power__less__imp__less__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3811_power__less__imp__less__base,axiom,
    ! [A: code_integer,Na: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3812_power__less__imp__less__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3813_power__less__imp__less__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3814_power__less__imp__less__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3815_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3816_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3817_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3818_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3819_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3820_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3821_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3822_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3823_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_3824_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_3825_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_3826_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_3827_zero__less__two,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,one_one_Code_integer)) ).

% zero_less_two
tff(fact_3828_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_3829_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3830_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_3831_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_3832_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_3833_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3834_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_3835_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3836_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_3837_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_3838_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_3839_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_3840_power__gt1__lemma,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))) ) ).

% power_gt1_lemma
tff(fact_3841_power__gt1__lemma,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_gt1_lemma
tff(fact_3842_power__gt1__lemma,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))) ) ).

% power_gt1_lemma
tff(fact_3843_power__gt1__lemma,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))) ) ).

% power_gt1_lemma
tff(fact_3844_power__gt1__lemma,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))) ) ).

% power_gt1_lemma
tff(fact_3845_power__less__power__Suc,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))) ) ).

% power_less_power_Suc
tff(fact_3846_power__less__power__Suc,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_less_power_Suc
tff(fact_3847_power__less__power__Suc,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))) ) ).

% power_less_power_Suc
tff(fact_3848_power__less__power__Suc,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))) ) ).

% power_less_power_Suc
tff(fact_3849_power__less__power__Suc,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))) ) ).

% power_less_power_Suc
tff(fact_3850_numeral__Bit1,axiom,
    ! [Na: num] : numera6690914467698888265omplex(aa_num_num(bit1,Na)) = plus_plus_complex(plus_plus_complex(numera6690914467698888265omplex(Na),numera6690914467698888265omplex(Na)),one_one_complex) ).

% numeral_Bit1
tff(fact_3851_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),aa_num_real(numeral_numeral_real,Na))),one_one_real) ).

% numeral_Bit1
tff(fact_3852_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_num_rat(numeral_numeral_rat,Na))),one_one_rat) ).

% numeral_Bit1
tff(fact_3853_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = 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,Na)),aa_num_nat(numeral_numeral_nat,Na))),one_one_nat) ).

% numeral_Bit1
tff(fact_3854_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)) = 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,Na)),aa_num_int(numeral_numeral_int,Na))),one_one_int) ).

% numeral_Bit1
tff(fact_3855_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,Na)),one_one_Code_integer) ).

% numeral_Bit1
tff(fact_3856_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = $ite(Na = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_3857_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Na) = $ite(Na = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_3858_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = $ite(Na = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_3859_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Na) = $ite(Na = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_3860_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),Na) = $ite(Na = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).

% power_0_left
tff(fact_3861_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Na) = $ite(Na = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_3862_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M)) ) ) ).

% div_le_mono2
tff(fact_3863_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% div_greater_zero_iff
tff(fact_3864_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_3865_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_3866_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_3867_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_3868_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_3869_power__strict__increasing,axiom,
    ! [Na: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)) ) ) ).

% power_strict_increasing
tff(fact_3870_power__strict__increasing,axiom,
    ! [Na: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N3)) ) ) ).

% power_strict_increasing
tff(fact_3871_power__strict__increasing,axiom,
    ! [Na: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)) ) ) ).

% power_strict_increasing
tff(fact_3872_power__strict__increasing,axiom,
    ! [Na: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N3)) ) ) ).

% power_strict_increasing
tff(fact_3873_power__strict__increasing,axiom,
    ! [Na: nat,N3: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N3)) ) ) ).

% power_strict_increasing
tff(fact_3874_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_3875_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),Na) ) ).

% mod_le_divisor
tff(fact_3876_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ~ ! [S2: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_3877_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ~ ! [S2: nat] : Na != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_3878_nat__mod__eq__lemma,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
       => ? [Q3: nat] : Xa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3879_ceiling__add__le,axiom,
    ! [Xa: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_real_int(archim7802044766580827645g_real,Y))) ).

% ceiling_add_le
tff(fact_3880_ceiling__add__le,axiom,
    ! [Xa: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_rat_int(archim2889992004027027881ng_rat,Y))) ).

% ceiling_add_le
tff(fact_3881_real__of__nat__div4,axiom,
    ! [Na: nat,Xa: 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,Na),Xa))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))) ).

% real_of_nat_div4
tff(fact_3882_of__nat__ceiling,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R)))) ).

% of_nat_ceiling
tff(fact_3883_of__nat__ceiling,axiom,
    ! [R: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R)))) ).

% of_nat_ceiling
tff(fact_3884_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_3885_reals__Archimedean3,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ! [Y4: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),Xa)) ) ).

% reals_Archimedean3
tff(fact_3886_zdiv__mult__self,axiom,
    ! [M: int,A: int,Na: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,M),Na))),M) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),M)),Na) ) ) ).

% zdiv_mult_self
tff(fact_3887_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),Na) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),Na) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_3888_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% le_log2_of_power
tff(fact_3889_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Na)) = $let(
        m: complex,
        m:= numera6690914467698888265omplex(Na),
        plus_plus_complex(plus_plus_complex(m,m),one_one_complex) ) ).

% numeral_code(3)
tff(fact_3890_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Na)) = $let(
        m: real,
        m:= aa_num_real(numeral_numeral_real,Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m)),one_one_real) ) ).

% numeral_code(3)
tff(fact_3891_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Na)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,Na),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m),m)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_3892_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,Na),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_3893_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Na),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m)),one_one_int) ) ).

% numeral_code(3)
tff(fact_3894_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,Na),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(m,m),one_one_Code_integer) ) ).

% numeral_code(3)
tff(fact_3895_real__nat__ceiling__ge,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xa)))) ).

% real_nat_ceiling_ge
tff(fact_3896_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_3897_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3898_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(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(A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_3899_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3900_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3901_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3902_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3903_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3904_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3905_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3906_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3907_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3908_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3909_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3910_mult__imp__div__pos__le,axiom,
    ! [Y: real,Xa: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(times_times_real(Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xa),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3911_mult__imp__div__pos__le,axiom,
    ! [Y: rat,Xa: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3912_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),Y)),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(divide_divide_real(Xa),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3913_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3914_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3915_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3916_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3917_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) = aa_nat_Code_integer(power_8256067586552552935nteger(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3918_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3919_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3920_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3921_power__eq__imp__eq__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3922_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,Na: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) = aa_nat_Code_integer(power_8256067586552552935nteger(B),Na) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3923_power__eq__imp__eq__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B),Na) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3924_power__eq__imp__eq__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3925_power__eq__imp__eq__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B),Na) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3926_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_3927_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_3928_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_3929_power__Suc__less,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% power_Suc_less
tff(fact_3930_power__Suc__less,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na))),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% power_Suc_less
tff(fact_3931_power__Suc__less,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% power_Suc_less
tff(fact_3932_power__Suc__less,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))),aa_nat_int(power_power_int(A),Na)) ) ) ).

% power_Suc_less
tff(fact_3933_power__Suc__less,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ) ).

% power_Suc_less
tff(fact_3934_power__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_3935_power__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_3936_power__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_3937_power__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_3938_power__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N3)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_3939_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_3940_one__power2,axiom,
    aa_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_3941_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_3942_one__power2,axiom,
    aa_nat_complex(power_power_complex(one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_3943_one__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_power2
tff(fact_3944_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_3945_one__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% one_less_power
tff(fact_3946_one__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% one_less_power
tff(fact_3947_one__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% one_less_power
tff(fact_3948_one__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ) ).

% one_less_power
tff(fact_3949_one__less__power,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ) ).

% one_less_power
tff(fact_3950_td__gal,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)) ) ) ).

% td_gal
tff(fact_3951_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Q2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3952_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B)))
         => ( plus_p5714425477246183910nteger(times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),divide6298287555418463151nteger(A,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3953_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3954_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3955_realpow__pos__nth,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R4),Na) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_3956_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na) = A )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),Na) = A ) )
               => ( Y4 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3957_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_3958_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_3959_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N2: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => aa_nat_o(P,N2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_3960_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_le
tff(fact_3961_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_3962_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),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(aa_rat_fun_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(aa_rat_fun_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_3963_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_3964_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(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_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(aa_rat_fun_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_3965_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3966_zero__le__power2,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3967_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3968_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3969_power2__eq__imp__eq,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( Xa = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3970_power2__eq__imp__eq,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
         => ( Xa = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3971_power2__eq__imp__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( Xa = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3972_power2__eq__imp__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( Xa = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3973_power2__eq__imp__eq,axiom,
    ! [Xa: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( Xa = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3974_power2__le__imp__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% power2_le_imp_le
tff(fact_3975_power2__le__imp__le,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y) ) ) ).

% power2_le_imp_le
tff(fact_3976_power2__le__imp__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ).

% power2_le_imp_le
tff(fact_3977_power2__le__imp__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power2_le_imp_le
tff(fact_3978_power2__le__imp__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ).

% power2_le_imp_le
tff(fact_3979_power__strict__mono,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_3980_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_3981_power__strict__mono,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_3982_power__strict__mono,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_3983_power__strict__mono,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_3984_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( modulo364778990260209775nteger(A,times_3573771949741848930nteger(B,C)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(B,modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3985_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3986_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3987_log__base__pow,axiom,
    ! [A: real,Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(log(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),Xa) = aa_real_real(divide_divide_real(aa_real_real(log(A),Xa)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log_base_pow
tff(fact_3988_log__nat__power,axiom,
    ! [Xa: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(log(B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(B),Xa)) ) ) ).

% log_nat_power
tff(fact_3989_power2__less__imp__less,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ).

% power2_less_imp_less
tff(fact_3990_power2__less__imp__less,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y) ) ) ).

% power2_less_imp_less
tff(fact_3991_power2__less__imp__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ) ).

% power2_less_imp_less
tff(fact_3992_power2__less__imp__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power2_less_imp_less
tff(fact_3993_power2__less__imp__less,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ) ).

% power2_less_imp_less
tff(fact_3994_sum__power2__ge__zero,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3995_sum__power2__ge__zero,axiom,
    ! [Xa: code_integer,Y: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3996_sum__power2__ge__zero,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3997_sum__power2__ge__zero,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3998_sum__power2__le__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3999_sum__power2__le__zero__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger)
    <=> ( ( Xa = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_4000_sum__power2__le__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_4001_sum__power2__le__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_4002_zero__le__even__power_H,axiom,
    ! [A: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_4003_zero__le__even__power_H,axiom,
    ! [A: code_integer,Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_4004_zero__le__even__power_H,axiom,
    ! [A: rat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_4005_zero__le__even__power_H,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_4006_two__pow__div__gt__le,axiom,
    ! [V: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,V),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% two_pow_div_gt_le
tff(fact_4007_z1pmod2,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% z1pmod2
tff(fact_4008_z1pdiv2,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = B ).

% z1pdiv2
tff(fact_4009_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = plus_plus_complex(B,A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_4010_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_4011_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_4012_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_4013_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_4014_crossproduct__eq,axiom,
    ! [W: real,Y: real,Xa: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(W),Y)),aa_real_real(times_times_real(Xa),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(W),Z)),aa_real_real(times_times_real(Xa),Y)) )
    <=> ( ( W = Xa )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4015_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,Xa: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y)) )
    <=> ( ( W = Xa )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4016_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,Xa: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),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,Xa),Y)) )
    <=> ( ( W = Xa )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4017_crossproduct__eq,axiom,
    ! [W: int,Y: int,Xa: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,Xa),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,Xa),Y)) )
    <=> ( ( W = Xa )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4018_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4019_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4020_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != 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,A),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,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4021_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != 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,A),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,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4022_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B5)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B5),A4) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),zero_zero_nat)
       => ( ! [A4: nat,B5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B5)
             => aa_nat_o(aa_nat_fun_nat_o(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B5)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_4023_sum__squares__bound,axiom,
    ! [Xa: real,Y: 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))),Xa)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_4024_sum__squares__bound,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xa)),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_4025_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(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(A),C))),aa_real_real(times_times_real(B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(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_4026_p1mod22k_H,axiom,
    ! [B: int,Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))) ).

% p1mod22k'
tff(fact_4027_p1mod22k,axiom,
    ! [B: int,Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))),one_one_int) ).

% p1mod22k
tff(fact_4028_mod__double__modulus,axiom,
    ! [M: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
       => ( ( modulo364778990260209775nteger(Xa,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),M)) = modulo364778990260209775nteger(Xa,M) )
          | ( modulo364778990260209775nteger(Xa,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Xa,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_4029_mod__double__modulus,axiom,
    ! [M: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_4030_mod__double__modulus,axiom,
    ! [M: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),M) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_4031_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4032_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4033_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4034_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4035_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4036_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4037_vebt__buildup__bound,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_4038_Tbuildupi__buildupi_H,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) = aa_nat_int(vEBT_V9176841429113362141ildupi,Na) ).

% Tbuildupi_buildupi'
tff(fact_4039_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4040_set__bit__0,axiom,
    ! [A: code_integer] : bit_se2793503036327961859nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(one_one_Code_integer,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4041_set__bit__0,axiom,
    ! [A: int] : bit_se7879613467334960850it_int(zero_zero_nat,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4042_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4043_two__realpow__ge__two,axiom,
    ! [Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)) ) ).

% two_realpow_ge_two
tff(fact_4044_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Na: 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,Na)),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),Na)),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,Na),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_4045_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)
     => ( ! [X: int] :
            ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X5: int] :
              ( aa_int_o(P,X5)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_4046_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Na: 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),Na)),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,Na),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,Na)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_4047_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => aa_set_real_o(member_real(X),B2) )
     => aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ).

% subsetI
tff(fact_4048_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => aa_set_int_o(member_int(X),B2) )
     => aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ).

% subsetI
tff(fact_4049_subsetI,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B2) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2) ) ).

% subsetI
tff(fact_4050_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => aa_set_nat_o(member_nat(X),B2) )
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% subsetI
tff(fact_4051_subset__antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
       => ( A2 = B2 ) ) ) ).

% subset_antisym
tff(fact_4052_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_4053_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_4054_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_4055_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_4056_power__one__right,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),one_one_nat) = A ).

% power_one_right
tff(fact_4057_power__one__right,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),one_one_nat) = A ).

% power_one_right
tff(fact_4058_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_4059_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_4060_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se7879613467334960850it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_4061_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se7879613467334960850it_int(Na,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_4062_less__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),one_one_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% less_one
tff(fact_4063_log__one,axiom,
    ! [A: real] : aa_real_real(log(A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_4064_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = M )
      <=> ( Na = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_4065_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(log(A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_4066_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),Xa)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_4067_one__less__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(log(A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_4068_log__less__one__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),Xa)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_4069_log__less__cancel__iff,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),Xa)),aa_real_real(log(A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_4070_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_4071_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(log(A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_4072_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),Xa)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_4073_one__le__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(log(A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xa) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_4074_log__le__one__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),Xa)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_4075_log__le__cancel__iff,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),Xa)),aa_real_real(log(A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_4076_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_4077_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_4078_iadd__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(plus_p3455044024723400733d_enat,M),Na) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( Na = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_4079_imult__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(times_7803423173614009249d_enat,M),Na) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( Na = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_4080_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Na),zero_z5237406670263579293d_enat)
    <=> ( Na = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_4081_i0__lb,axiom,
    ! [Na: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Na) ).

% i0_lb
tff(fact_4082_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_4083_pred__subset__eq,axiom,
    ! [R2: set_real,S: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_a(R2),aTP_Lamm_a(S))
    <=> aa_set_real_o(ord_less_eq_set_real(R2),S) ) ).

% pred_subset_eq
tff(fact_4084_pred__subset__eq,axiom,
    ! [R2: set_int,S: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ac(R2),aTP_Lamm_ac(S))
    <=> aa_set_int_o(ord_less_eq_set_int(R2),S) ) ).

% pred_subset_eq
tff(fact_4085_pred__subset__eq,axiom,
    ! [R2: set_VEBT_VEBT,S: set_VEBT_VEBT] :
      ( ord_le88303844413636359VEBT_o(aTP_Lamm_aa(R2),aTP_Lamm_aa(S))
    <=> aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(R2),S) ) ).

% pred_subset_eq
tff(fact_4086_pred__subset__eq,axiom,
    ! [R2: set_nat,S: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ab(R2),aTP_Lamm_ab(S))
    <=> aa_set_nat_o(ord_less_eq_set_nat(R2),S) ) ).

% pred_subset_eq
tff(fact_4087_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_4088_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se4203085406695923979it_int(Na,K)),K) ).

% unset_bit_less_eq
tff(fact_4089_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),bit_se7879613467334960850it_int(Na,K)) ).

% set_bit_greater_eq
tff(fact_4090_in__mono,axiom,
    ! [A2: set_real,B2: set_real,Xa: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => aa_set_real_o(member_real(Xa),B2) ) ) ).

% in_mono
tff(fact_4091_in__mono,axiom,
    ! [A2: set_int,B2: set_int,Xa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => aa_set_int_o(member_int(Xa),B2) ) ) ).

% in_mono
tff(fact_4092_in__mono,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B2) ) ) ).

% in_mono
tff(fact_4093_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,Xa: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => aa_set_nat_o(member_nat(Xa),B2) ) ) ).

% in_mono
tff(fact_4094_subsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B2) ) ) ).

% subsetD
tff(fact_4095_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% subsetD
tff(fact_4096_subsetD,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ) ).

% subsetD
tff(fact_4097_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% subsetD
tff(fact_4098_equalityE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% equalityE
tff(fact_4099_subset__eq,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),A2)
         => aa_set_real_o(member_real(X4),B2) ) ) ).

% subset_eq
tff(fact_4100_subset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A2)
         => aa_set_int_o(member_int(X4),B2) ) ) ).

% subset_eq
tff(fact_4101_subset__eq,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2)
    <=> ! [X4: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),A2)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),B2) ) ) ).

% subset_eq
tff(fact_4102_subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A2)
         => aa_set_nat_o(member_nat(X4),B2) ) ) ).

% subset_eq
tff(fact_4103_equalityD1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% equalityD1
tff(fact_4104_Set_OequalityD2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ).

% Set.equalityD2
tff(fact_4105_subset__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ! [T2: real] :
          ( aa_set_real_o(member_real(T2),A2)
         => aa_set_real_o(member_real(T2),B2) ) ) ).

% subset_iff
tff(fact_4106_subset__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ! [T2: int] :
          ( aa_set_int_o(member_int(T2),A2)
         => aa_set_int_o(member_int(T2),B2) ) ) ).

% subset_iff
tff(fact_4107_subset__iff,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2)
    <=> ! [T2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T2),A2)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T2),B2) ) ) ).

% subset_iff
tff(fact_4108_subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ! [T2: nat] :
          ( aa_set_nat_o(member_nat(T2),A2)
         => aa_set_nat_o(member_nat(T2),B2) ) ) ).

% subset_iff
tff(fact_4109_subset__refl,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A2),A2) ).

% subset_refl
tff(fact_4110_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( aa_int_o(P,X)
         => aa_int_o(Q,X) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q)) ) ).

% Collect_mono
tff(fact_4111_Collect__mono,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X: complex] :
          ( aa_complex_o(P,X)
         => aa_complex_o(Q,X) )
     => aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q)) ) ).

% Collect_mono
tff(fact_4112_Collect__mono,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ! [X: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X)
         => aa_Pro4592068669816069595_int_o(Q,X) )
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q)) ) ).

% Collect_mono
tff(fact_4113_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(P,X)
         => aa_nat_o(Q,X) )
     => aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_4114_subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
       => aa_set_nat_o(ord_less_eq_set_nat(A2),C2) ) ) ).

% subset_trans
tff(fact_4115_set__eq__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% set_eq_subset
tff(fact_4116_less__eq__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_eq_set_def
tff(fact_4117_less__eq__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_eq_set_def
tff(fact_4118_less__eq__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2)
    <=> ord_le88303844413636359VEBT_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_eq_set_def
tff(fact_4119_less__eq__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2)) ) ).

% less_eq_set_def
tff(fact_4120_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))
    <=> ! [X4: int] :
          ( aa_int_o(P,X4)
         => aa_int_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_4121_Collect__mono__iff,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q))
    <=> ! [X4: complex] :
          ( aa_complex_o(P,X4)
         => aa_complex_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_4122_Collect__mono__iff,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q))
    <=> ! [X4: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X4)
         => aa_Pro4592068669816069595_int_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_4123_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))
    <=> ! [X4: nat] :
          ( aa_nat_o(P,X4)
         => aa_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_4124_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X5: real] : aa_set_real_o(member_real(X5),S)
     => ( ? [Z5: real] :
          ! [X: real] :
            ( aa_set_real_o(member_real(X),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z5) )
       => ? [Y2: real] :
            ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),Y2) )
            & ! [Z5: real] :
                ( ! [X: real] :
                    ( aa_set_real_o(member_real(X),S)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z5) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),Z5) ) ) ) ) ).

% complete_real
tff(fact_4125_le__refl,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Na) ).

% le_refl
tff(fact_4126_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_4127_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% eq_imp_le
tff(fact_4128_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_4129_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% nat_le_linear
tff(fact_4130_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B) )
       => ? [X: nat] :
            ( aa_nat_o(P,X)
            & ! [Y4: nat] :
                ( aa_nat_o(P,Y4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_4131_verit__la__generic,axiom,
    ! [A: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
      | ( A = Xa )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A) ) ).

% verit_la_generic
tff(fact_4132_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ci(A2),P))),A2) ).

% Collect_subset
tff(fact_4133_Collect__subset,axiom,
    ! [A2: set_VEBT_VEBT,P: fun_VEBT_VEBT_o] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_cj(A2),P))),A2) ).

% Collect_subset
tff(fact_4134_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ck(A2),P))),A2) ).

% Collect_subset
tff(fact_4135_Collect__subset,axiom,
    ! [A2: set_complex,P: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_cl(A2),P))),A2) ).

% Collect_subset
tff(fact_4136_Collect__subset,axiom,
    ! [A2: set_Pr958786334691620121nt_int,P: fun_Pr3238374608701176162_int_o] : ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_cm(A2),P)),A2) ).

% Collect_subset
tff(fact_4137_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cn(A2),P))),A2) ).

% Collect_subset
tff(fact_4138_nat__mult__1__right,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),one_one_nat) = Na ).

% nat_mult_1_right
tff(fact_4139_nat__mult__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Na) = Na ).

% nat_mult_1
tff(fact_4140_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_4141_mult__eq__self__implies__10,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
     => ( ( Na = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_4142_real__arch__pow,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),N)) ) ).

% real_arch_pow
tff(fact_4143_real__arch__pow__inv,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_4144_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_4145_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_4146_pinf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z5: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z5),X)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z5: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z5),X)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q6,X) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
           => ( ( aa_real_o(P,X5)
                & aa_real_o(Q,X5) )
            <=> ( aa_real_o(P2,X5)
                & aa_real_o(Q6,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_4147_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z5: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q6,X) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
           => ( ( aa_rat_o(P,X5)
                & aa_rat_o(Q,X5) )
            <=> ( aa_rat_o(P2,X5)
                & aa_rat_o(Q6,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_4148_pinf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z5: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z5: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q6,X) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X5)
           => ( ( aa_num_o(P,X5)
                & aa_num_o(Q,X5) )
            <=> ( aa_num_o(P2,X5)
                & aa_num_o(Q6,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_4149_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q6,X) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
           => ( ( aa_nat_o(P,X5)
                & aa_nat_o(Q,X5) )
            <=> ( aa_nat_o(P2,X5)
                & aa_nat_o(Q6,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_4150_pinf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z5: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z5: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q6,X) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
           => ( ( aa_int_o(P,X5)
                & aa_int_o(Q,X5) )
            <=> ( aa_int_o(P2,X5)
                & aa_int_o(Q6,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_4151_pinf_I1_J,axiom,
    ! [P: fun_Code_integer_o,P2: fun_Code_integer_o,Q: fun_Code_integer_o,Q6: fun_Code_integer_o] :
      ( ? [Z5: code_integer] :
        ! [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z5),X)
         => ( aa_Code_integer_o(P,X)
          <=> aa_Code_integer_o(P2,X) ) )
     => ( ? [Z5: code_integer] :
          ! [X: code_integer] :
            ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z5),X)
           => ( aa_Code_integer_o(Q,X)
            <=> aa_Code_integer_o(Q6,X) ) )
       => ? [Z3: code_integer] :
          ! [X5: code_integer] :
            ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
           => ( ( aa_Code_integer_o(P,X5)
                & aa_Code_integer_o(Q,X5) )
            <=> ( aa_Code_integer_o(P2,X5)
                & aa_Code_integer_o(Q6,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_4152_pinf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z5: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z5),X)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z5: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z5),X)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q6,X) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
           => ( ( aa_real_o(P,X5)
                | aa_real_o(Q,X5) )
            <=> ( aa_real_o(P2,X5)
                | aa_real_o(Q6,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_4153_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z5: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q6,X) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
           => ( ( aa_rat_o(P,X5)
                | aa_rat_o(Q,X5) )
            <=> ( aa_rat_o(P2,X5)
                | aa_rat_o(Q6,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_4154_pinf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z5: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z5: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q6,X) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X5)
           => ( ( aa_num_o(P,X5)
                | aa_num_o(Q,X5) )
            <=> ( aa_num_o(P2,X5)
                | aa_num_o(Q6,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_4155_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q6,X) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
           => ( ( aa_nat_o(P,X5)
                | aa_nat_o(Q,X5) )
            <=> ( aa_nat_o(P2,X5)
                | aa_nat_o(Q6,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_4156_pinf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z5: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z5: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q6,X) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
           => ( ( aa_int_o(P,X5)
                | aa_int_o(Q,X5) )
            <=> ( aa_int_o(P2,X5)
                | aa_int_o(Q6,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_4157_pinf_I2_J,axiom,
    ! [P: fun_Code_integer_o,P2: fun_Code_integer_o,Q: fun_Code_integer_o,Q6: fun_Code_integer_o] :
      ( ? [Z5: code_integer] :
        ! [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z5),X)
         => ( aa_Code_integer_o(P,X)
          <=> aa_Code_integer_o(P2,X) ) )
     => ( ? [Z5: code_integer] :
          ! [X: code_integer] :
            ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z5),X)
           => ( aa_Code_integer_o(Q,X)
            <=> aa_Code_integer_o(Q6,X) ) )
       => ? [Z3: code_integer] :
          ! [X5: code_integer] :
            ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
           => ( ( aa_Code_integer_o(P,X5)
                | aa_Code_integer_o(Q,X5) )
            <=> ( aa_Code_integer_o(P2,X5)
                | aa_Code_integer_o(Q6,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_4158_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_4159_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_4160_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_4161_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_4162_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_4163_pinf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_4164_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_4165_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_4166_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_4167_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_4168_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_4169_pinf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_4170_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X5),T) ) ).

% pinf(5)
tff(fact_4171_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),T) ) ).

% pinf(5)
tff(fact_4172_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X5)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X5),T) ) ).

% pinf(5)
tff(fact_4173_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),T) ) ).

% pinf(5)
tff(fact_4174_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X5),T) ) ).

% pinf(5)
tff(fact_4175_pinf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),T) ) ).

% pinf(5)
tff(fact_4176_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,T),X5) ) ).

% pinf(7)
tff(fact_4177_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X5) ) ).

% pinf(7)
tff(fact_4178_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X5)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,T),X5) ) ).

% pinf(7)
tff(fact_4179_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X5) ) ).

% pinf(7)
tff(fact_4180_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,T),X5) ) ).

% pinf(7)
tff(fact_4181_pinf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,T),X5) ) ).

% pinf(7)
tff(fact_4182_minf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z5: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z5)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z5: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z5)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q6,X) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
           => ( ( aa_real_o(P,X5)
                & aa_real_o(Q,X5) )
            <=> ( aa_real_o(P2,X5)
                & aa_real_o(Q6,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_4183_minf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z5)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z5: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z5)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q6,X) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
           => ( ( aa_rat_o(P,X5)
                & aa_rat_o(Q,X5) )
            <=> ( aa_rat_o(P2,X5)
                & aa_rat_o(Q6,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_4184_minf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z5: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z5)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z5: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z5)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q6,X) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z3)
           => ( ( aa_num_o(P,X5)
                & aa_num_o(Q,X5) )
            <=> ( aa_num_o(P2,X5)
                & aa_num_o(Q6,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_4185_minf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z5)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z5)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q6,X) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
           => ( ( aa_nat_o(P,X5)
                & aa_nat_o(Q,X5) )
            <=> ( aa_nat_o(P2,X5)
                & aa_nat_o(Q6,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_4186_minf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z5: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z5)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z5: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z5)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q6,X) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
           => ( ( aa_int_o(P,X5)
                & aa_int_o(Q,X5) )
            <=> ( aa_int_o(P2,X5)
                & aa_int_o(Q6,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_4187_minf_I1_J,axiom,
    ! [P: fun_Code_integer_o,P2: fun_Code_integer_o,Q: fun_Code_integer_o,Q6: fun_Code_integer_o] :
      ( ? [Z5: code_integer] :
        ! [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z5)
         => ( aa_Code_integer_o(P,X)
          <=> aa_Code_integer_o(P2,X) ) )
     => ( ? [Z5: code_integer] :
          ! [X: code_integer] :
            ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z5)
           => ( aa_Code_integer_o(Q,X)
            <=> aa_Code_integer_o(Q6,X) ) )
       => ? [Z3: code_integer] :
          ! [X5: code_integer] :
            ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
           => ( ( aa_Code_integer_o(P,X5)
                & aa_Code_integer_o(Q,X5) )
            <=> ( aa_Code_integer_o(P2,X5)
                & aa_Code_integer_o(Q6,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_4188_minf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z5: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z5)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z5: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z5)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q6,X) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
           => ( ( aa_real_o(P,X5)
                | aa_real_o(Q,X5) )
            <=> ( aa_real_o(P2,X5)
                | aa_real_o(Q6,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_4189_minf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z5)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z5: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z5)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q6,X) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
           => ( ( aa_rat_o(P,X5)
                | aa_rat_o(Q,X5) )
            <=> ( aa_rat_o(P2,X5)
                | aa_rat_o(Q6,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_4190_minf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z5: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z5)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z5: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z5)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q6,X) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z3)
           => ( ( aa_num_o(P,X5)
                | aa_num_o(Q,X5) )
            <=> ( aa_num_o(P2,X5)
                | aa_num_o(Q6,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_4191_minf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z5)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z5)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q6,X) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
           => ( ( aa_nat_o(P,X5)
                | aa_nat_o(Q,X5) )
            <=> ( aa_nat_o(P2,X5)
                | aa_nat_o(Q6,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_4192_minf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z5: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z5)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z5: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z5)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q6,X) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
           => ( ( aa_int_o(P,X5)
                | aa_int_o(Q,X5) )
            <=> ( aa_int_o(P2,X5)
                | aa_int_o(Q6,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_4193_minf_I2_J,axiom,
    ! [P: fun_Code_integer_o,P2: fun_Code_integer_o,Q: fun_Code_integer_o,Q6: fun_Code_integer_o] :
      ( ? [Z5: code_integer] :
        ! [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z5)
         => ( aa_Code_integer_o(P,X)
          <=> aa_Code_integer_o(P2,X) ) )
     => ( ? [Z5: code_integer] :
          ! [X: code_integer] :
            ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z5)
           => ( aa_Code_integer_o(Q,X)
            <=> aa_Code_integer_o(Q6,X) ) )
       => ? [Z3: code_integer] :
          ! [X5: code_integer] :
            ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
           => ( ( aa_Code_integer_o(P,X5)
                | aa_Code_integer_o(Q,X5) )
            <=> ( aa_Code_integer_o(P2,X5)
                | aa_Code_integer_o(Q6,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_4194_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_4195_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_4196_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z3)
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_4197_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_4198_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_4199_minf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_4200_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_4201_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_4202_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z3)
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_4203_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_4204_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_4205_minf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_4206_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X5),T) ) ).

% minf(5)
tff(fact_4207_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),T) ) ).

% minf(5)
tff(fact_4208_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X5),T) ) ).

% minf(5)
tff(fact_4209_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),T) ) ).

% minf(5)
tff(fact_4210_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X5),T) ) ).

% minf(5)
tff(fact_4211_minf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),T) ) ).

% minf(5)
tff(fact_4212_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,T),X5) ) ).

% minf(7)
tff(fact_4213_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X5) ) ).

% minf(7)
tff(fact_4214_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,T),X5) ) ).

% minf(7)
tff(fact_4215_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X5) ) ).

% minf(7)
tff(fact_4216_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,T),X5) ) ).

% minf(7)
tff(fact_4217_minf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,T),X5) ) ).

% minf(7)
tff(fact_4218_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),M) ) ) ).

% div_less_dividend
tff(fact_4219_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M)) ) ).

% nat_less_real_le
tff(fact_4220_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_4221_log__base__change,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(B),Xa) = aa_real_real(divide_divide_real(aa_real_real(log(A),Xa)),aa_real_real(log(A),B)) ) ) ) ).

% log_base_change
tff(fact_4222_nat__induct2,axiom,
    ! [P: fun_nat_o,Na: 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,Na) ) ) ) ).

% nat_induct2
tff(fact_4223_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_4224_two__realpow__ge__one,axiom,
    ! [Na: 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))),Na)) ).

% two_realpow_ge_one
tff(fact_4225_log__mult,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(log(A),aa_real_real(times_times_real(Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(A),Xa)),aa_real_real(log(A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_4226_less__log__of__power,axiom,
    ! [B: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(B),M)) ) ) ).

% less_log_of_power
tff(fact_4227_log__of__power__eq,axiom,
    ! [M: nat,B: real,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_4228_le__log__of__power,axiom,
    ! [B: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(B),M)) ) ) ).

% le_log_of_power
tff(fact_4229_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Na) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Na)) ) ) ).

% verit_le_mono_div
tff(fact_4230_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_4231_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_4232_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),T) ) ).

% pinf(6)
tff(fact_4233_pinf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X5),T) ) ).

% pinf(6)
tff(fact_4234_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),T) ) ).

% pinf(6)
tff(fact_4235_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X5)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),T) ) ).

% pinf(6)
tff(fact_4236_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),T) ) ).

% pinf(6)
tff(fact_4237_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),T) ) ).

% pinf(6)
tff(fact_4238_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X5) ) ).

% pinf(8)
tff(fact_4239_pinf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,T),X5) ) ).

% pinf(8)
tff(fact_4240_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X5) ) ).

% pinf(8)
tff(fact_4241_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X5)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T),X5) ) ).

% pinf(8)
tff(fact_4242_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X5) ) ).

% pinf(8)
tff(fact_4243_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X5) ) ).

% pinf(8)
tff(fact_4244_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),T) ) ).

% minf(6)
tff(fact_4245_minf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X5),T) ) ).

% minf(6)
tff(fact_4246_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),T) ) ).

% minf(6)
tff(fact_4247_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),T) ) ).

% minf(6)
tff(fact_4248_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),T) ) ).

% minf(6)
tff(fact_4249_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),T) ) ).

% minf(6)
tff(fact_4250_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X5) ) ).

% minf(8)
tff(fact_4251_minf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,T),X5) ) ).

% minf(8)
tff(fact_4252_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X5) ) ).

% minf(8)
tff(fact_4253_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T),X5) ) ).

% minf(8)
tff(fact_4254_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X5) ) ).

% minf(8)
tff(fact_4255_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X5) ) ).

% minf(8)
tff(fact_4256_log__of__power__less,axiom,
    ! [M: nat,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_less
tff(fact_4257_log__of__power__le,axiom,
    ! [M: nat,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_le
tff(fact_4258_power__2__mult__step__le,axiom,
    ! [N6: nat,Na: nat,K4: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)),K4)),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))),Na)),K))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K4),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_4259_linear__plus__1__le__power,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),Xa)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),one_one_real)),Na)) ) ).

% linear_plus_1_le_power
tff(fact_4260_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_4261_buildup__build__time,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Na)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Na)) ).

% buildup_build_time
tff(fact_4262_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),vEBT_VEBT_cnt(T)) ).

% cnt_non_neg
tff(fact_4263_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_4264_log__ceil__idem,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)) = 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,Xa)))) ) ) ).

% log_ceil_idem
tff(fact_4265_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( 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,Na))) = 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,Na),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_4266_ceiling__log__eq__powr__iff,axiom,
    ! [Xa: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( 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),Xa)) = 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))),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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_4267_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,W) = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( W = Z ) ) ).

% of_int_eq_iff
tff(fact_4268_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,W) = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( W = Z ) ) ).

% of_int_eq_iff
tff(fact_4269_diff__self,axiom,
    ! [A: complex] : minus_minus_complex(A,A) = zero_zero_complex ).

% diff_self
tff(fact_4270_diff__self,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% diff_self
tff(fact_4271_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% diff_self
tff(fact_4272_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_4273_diff__0__right,axiom,
    ! [A: complex] : minus_minus_complex(A,zero_zero_complex) = A ).

% diff_0_right
tff(fact_4274_diff__0__right,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_0_right
tff(fact_4275_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_4276_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_4277_zero__diff,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_diff
tff(fact_4278_diff__zero,axiom,
    ! [A: complex] : minus_minus_complex(A,zero_zero_complex) = A ).

% diff_zero
tff(fact_4279_diff__zero,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_zero
tff(fact_4280_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_zero
tff(fact_4281_diff__zero,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).

% diff_zero
tff(fact_4282_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_4283_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : minus_minus_complex(A,A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4284_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4285_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4286_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4287_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4288_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_4289_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel
tff(fact_4290_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_4291_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A),B)),B) = A ).

% diff_add_cancel
tff(fact_4292_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ).

% diff_add_cancel
tff(fact_4293_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_4294_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_left
tff(fact_4295_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_left
tff(fact_4296_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_4297_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_4298_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_4299_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_4300_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_4301_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_4302_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_right
tff(fact_4303_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_right
tff(fact_4304_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_4305_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_4306_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_4307_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_4308_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_4309_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_4310_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mod_self2
tff(fact_4311_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% minus_mod_self2
tff(fact_4312_diff__0__eq__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Na) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_4313_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_4314_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_4315_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_4316_of__int__diff,axiom,
    ! [W: int,Z: int] : ring_1_of_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,ring_1_of_int_int(W)),ring_1_of_int_int(Z)) ).

% of_int_diff
tff(fact_4317_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_4318_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_4319_powr__0,axiom,
    ! [Z: real] : aa_real_real(powr_real(zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_4320_powr__eq__0__iff,axiom,
    ! [W: real,Z: real] :
      ( ( aa_real_real(powr_real(W),Z) = zero_zero_real )
    <=> ( W = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_4321_powr__one__eq__one,axiom,
    ! [A: real] : aa_real_real(powr_real(one_one_real),A) = one_one_real ).

% powr_one_eq_one
tff(fact_4322_of__int__ceiling__cancel,axiom,
    ! [Xa: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xa)) = Xa )
    <=> ? [N2: int] : Xa = aa_int_real(ring_1_of_int_real,N2) ) ).

% of_int_ceiling_cancel
tff(fact_4323_of__int__ceiling__cancel,axiom,
    ! [Xa: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xa)) = Xa )
    <=> ? [N2: int] : Xa = aa_int_rat(ring_1_of_int_rat,N2) ) ).

% of_int_ceiling_cancel
tff(fact_4324_ceiling__of__int,axiom,
    ! [Z: int] : aa_real_int(archim7802044766580827645g_real,aa_int_real(ring_1_of_int_real,Z)) = Z ).

% ceiling_of_int
tff(fact_4325_ceiling__of__int,axiom,
    ! [Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_int_rat(ring_1_of_int_rat,Z)) = Z ).

% ceiling_of_int
tff(fact_4326_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_4327_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_4328_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_4329_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_4330_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_4331_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_4332_diff__gt__0__iff__gt,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_4333_zero__comp__diff__simps_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_4334_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_4335_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_4336_zero__comp__diff__simps_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_4337_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_4338_diff__numeral__special_I9_J,axiom,
    minus_minus_complex(one_one_complex,one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_4339_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_4340_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_4341_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_4342_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4343_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4344_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4345_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4346_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4347_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4348_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4349_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4350_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(minus_minus_complex(A,B)),numera6690914467698888265omplex(V)) = minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_4351_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),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_4352_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_4353_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_4354_left__diff__distrib__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B),aa_num_Code_integer(numera6620942414471956472nteger,V)) = minus_8373710615458151222nteger(times_3573771949741848930nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,V)),times_3573771949741848930nteger(B,aa_num_Code_integer(numera6620942414471956472nteger,V))) ).

% left_diff_distrib_numeral
tff(fact_4355_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),minus_minus_complex(B,C)) = minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_4356_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_4357_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_4358_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_4359_right__diff__distrib__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] : times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,V),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,V),B),times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,V),C)) ).

% right_diff_distrib_numeral
tff(fact_4360_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% zero_less_diff
tff(fact_4361_of__int__0,axiom,
    ring_17405671764205052669omplex(zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_4362_of__int__0,axiom,
    ring_1_of_int_int(zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_4363_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_4364_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_4365_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex = ring_17405671764205052669omplex(Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4366_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = ring_1_of_int_int(Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4367_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4368_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4369_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ring_17405671764205052669omplex(Z) = zero_zero_complex )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4370_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_int(Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4371_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = zero_zero_real )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4372_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4373_of__int__numeral,axiom,
    ! [K: num] : ring_17405671764205052669omplex(aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).

% of_int_numeral
tff(fact_4374_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_4375_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_4376_of__int__numeral,axiom,
    ! [K: num] : ring_1_of_int_int(aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_4377_of__int__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_num_int(numeral_numeral_int,K)) = aa_num_Code_integer(numera6620942414471956472nteger,K) ).

% of_int_numeral
tff(fact_4378_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( ring_17405671764205052669omplex(Z) = numera6690914467698888265omplex(Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4379_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4380_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4381_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( ring_1_of_int_int(Z) = aa_num_int(numeral_numeral_int,Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4382_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = aa_num_Code_integer(numera6620942414471956472nteger,Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4383_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% diff_is_0_eq
tff(fact_4384_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_4385_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_4386_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_4387_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(W)),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_4388_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_4389_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_4390_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_4391_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_4392_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_4393_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(W)),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_4394_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_4395_of__int__1,axiom,
    ring_1_of_int_int(one_one_int) = one_one_int ).

% of_int_1
tff(fact_4396_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_4397_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_4398_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_int(Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4399_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_4400_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_4401_powr__zero__eq__one,axiom,
    ! [Xa: real] :
      aa_real_real(powr_real(Xa),zero_zero_real) = $ite(Xa = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_4402_of__int__add,axiom,
    ! [W: int,Z: 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,ring_1_of_int_int(W)),ring_1_of_int_int(Z)) ).

% of_int_add
tff(fact_4403_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_4404_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(aa_rat_fun_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_4405_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_4406_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_4407_of__int__mult,axiom,
    ! [W: int,Z: 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,ring_1_of_int_int(W)),ring_1_of_int_int(Z)) ).

% of_int_mult
tff(fact_4408_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% of_int_of_nat_eq
tff(fact_4409_of__int__of__nat__eq,axiom,
    ! [Na: nat] : ring_1_of_int_int(aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% of_int_of_nat_eq
tff(fact_4410_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_rat(semiri681578069525770553at_rat,Na) ).

% of_int_of_nat_eq
tff(fact_4411_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) ).

% of_int_of_nat_eq
tff(fact_4412_of__int__of__nat__eq,axiom,
    ! [Na: nat] : ring_17405671764205052669omplex(aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_complex(semiri8010041392384452111omplex,Na) ).

% of_int_of_nat_eq
tff(fact_4413_of__int__power,axiom,
    ! [Z: int,Na: nat] : ring_1_of_int_int(aa_nat_int(power_power_int(Z),Na)) = aa_nat_int(power_power_int(ring_1_of_int_int(Z)),Na) ).

% of_int_power
tff(fact_4414_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Na) ).

% of_int_power
tff(fact_4415_of__int__power,axiom,
    ! [Z: int,Na: nat] : ring_17405671764205052669omplex(aa_nat_int(power_power_int(Z),Na)) = aa_nat_complex(power_power_complex(ring_17405671764205052669omplex(Z)),Na) ).

% of_int_power
tff(fact_4416_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(power_power_int(Z),Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Z)),Na) ).

% of_int_power
tff(fact_4417_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),Na)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Na) ).

% of_int_power
tff(fact_4418_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( ( aa_nat_int(power_power_int(ring_1_of_int_int(B)),W) = ring_1_of_int_int(Xa) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4419_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: 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,Xa) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4420_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( ( aa_nat_complex(power_power_complex(ring_17405671764205052669omplex(B)),W) = ring_17405671764205052669omplex(Xa) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4421_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W) = aa_int_Code_integer(ring_18347121197199848620nteger,Xa) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4422_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: 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,Xa) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4423_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( ring_1_of_int_int(Xa) = aa_nat_int(power_power_int(ring_1_of_int_int(B)),W) )
    <=> ( Xa = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4424_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xa) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( Xa = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4425_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( ring_17405671764205052669omplex(Xa) = aa_nat_complex(power_power_complex(ring_17405671764205052669omplex(B)),W) )
    <=> ( Xa = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4426_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Xa) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W) )
    <=> ( Xa = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4427_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Xa) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( Xa = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4428_ceiling__diff__of__int,axiom,
    ! [Xa: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(Xa),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xa)),Z) ).

% ceiling_diff_of_int
tff(fact_4429_ceiling__diff__of__int,axiom,
    ! [Xa: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),Z) ).

% ceiling_diff_of_int
tff(fact_4430_powr__gt__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(powr_real(Xa),A))
    <=> ( Xa != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_4431_powr__nonneg__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(A),Xa)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_4432_powr__less__cancel__iff,axiom,
    ! [Xa: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel_iff
tff(fact_4433_ceiling__diff__numeral,axiom,
    ! [Xa: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(Xa),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4434_ceiling__diff__numeral,axiom,
    ! [Xa: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4435_ceiling__diff__one,axiom,
    ! [Xa: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(Xa),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xa)),one_one_int) ).

% ceiling_diff_one
tff(fact_4436_ceiling__diff__one,axiom,
    ! [Xa: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),one_one_int) ).

% ceiling_diff_one
tff(fact_4437_ceiling__add__of__int,axiom,
    ! [Xa: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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,Xa)),Z) ).

% ceiling_add_of_int
tff(fact_4438_ceiling__add__of__int,axiom,
    ! [Xa: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),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,Xa)),Z) ).

% ceiling_add_of_int
tff(fact_4439_powr__eq__one__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(powr_real(A),Xa) = one_one_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_4440_powr__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),one_one_real) = Xa ) ) ).

% powr_one
tff(fact_4441_powr__one__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(powr_real(Xa),one_one_real) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% powr_one_gt_zero_iff
tff(fact_4442_powr__le__cancel__iff,axiom,
    ! [Xa: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_4443_numeral__powr__numeral__real,axiom,
    ! [M: num,Na: num] : aa_real_real(powr_real(aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,M)),aa_num_nat(numeral_numeral_nat,Na)) ).

% numeral_powr_numeral_real
tff(fact_4444_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_4445_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_4446_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_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_4447_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_4448_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_4449_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_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_4450_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),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,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4451_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4452_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),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,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4453_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4454_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: 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,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_4455_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_4456_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: 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,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_4457_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(Z)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_4458_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_4459_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_4460_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_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_4461_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_4462_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_4463_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_4464_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_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_4465_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_4466_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: 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,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_4467_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: 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,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_4468_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(Z)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_4469_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_4470_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),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,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4471_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),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,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4472_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4473_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4474_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_4475_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_4476_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_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_4477_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_4478_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_4479_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_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_4480_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_4481_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_4482_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_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_4483_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_4484_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_4485_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_4486_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_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_4487_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_4488_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xa)),Na) = ring_17405671764205052669omplex(Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4489_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4490_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4491_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) = ring_1_of_int_int(Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4492_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),Na) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4493_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( ring_17405671764205052669omplex(Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xa)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4494_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4495_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4496_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( ring_1_of_int_int(Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4497_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4498_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: 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,Xa)),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,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4499_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4500_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: 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,Xa)),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,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4501_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(Xa)),aa_nat_int(power_power_int(ring_1_of_int_int(B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4502_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: 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,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4503_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4504_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: 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,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4505_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(ring_1_of_int_int(B)),W)),ring_1_of_int_int(Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4506_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: 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,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4507_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: 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,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4508_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(ring_1_of_int_int(B)),W)),ring_1_of_int_int(Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4509_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4510_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xa)),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,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4511_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xa)),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,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4512_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(Xa)),aa_nat_int(power_power_int(ring_1_of_int_int(B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4513_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4514_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,Z)) = aa_int_real(ring_1_of_int_real,Z) ) ) ).

% of_nat_nat
tff(fact_4515_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = ring_1_of_int_int(Z) ) ) ).

% of_nat_nat
tff(fact_4516_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,Z)) = aa_int_rat(ring_1_of_int_rat,Z) ) ) ).

% of_nat_nat
tff(fact_4517_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,Z)) = aa_int_Code_integer(ring_18347121197199848620nteger,Z) ) ) ).

% of_nat_nat
tff(fact_4518_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,Z)) = ring_17405671764205052669omplex(Z) ) ) ).

% of_nat_nat
tff(fact_4519_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_real_real(powr_real(A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_4520_powr__log__cancel,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(powr_real(A),aa_real_real(log(A),Xa)) = Xa ) ) ) ) ).

% powr_log_cancel
tff(fact_4521_powr__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_num_real(numeral_numeral_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,Na)) ) ) ).

% powr_numeral
tff(fact_4522_ceiling__less__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_4523_ceiling__less__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_4524_numeral__le__ceiling,axiom,
    ! [V: num,Xa: 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,Xa))
    <=> 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)),Xa) ) ).

% numeral_le_ceiling
tff(fact_4525_numeral__le__ceiling,axiom,
    ! [V: num,Xa: 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,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xa) ) ).

% numeral_le_ceiling
tff(fact_4526_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4527_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4528_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4529_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),ring_1_of_int_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4530_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4531_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4532_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4533_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4534_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4535_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4536_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),ring_1_of_int_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4537_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4538_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4539_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4540_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4541_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4542_powr__diff,axiom,
    ! [W: real,Z1: real,Z2: real] : aa_real_real(powr_real(W),aa_real_real(minus_minus_real(Z1),Z2)) = aa_real_real(divide_divide_real(aa_real_real(powr_real(W),Z1)),aa_real_real(powr_real(W),Z2)) ).

% powr_diff
tff(fact_4543_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_4544_powr__powr__swap,axiom,
    ! [Xa: real,A: real,B: real] : aa_real_real(powr_real(aa_real_real(powr_real(Xa),A)),B) = aa_real_real(powr_real(aa_real_real(powr_real(Xa),B)),A) ).

% powr_powr_swap
tff(fact_4545_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4546_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4547_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4548_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_4549_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_4550_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_4551_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_4552_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% of_nat_diff
tff(fact_4553_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ) ).

% of_nat_diff
tff(fact_4554_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ) ).

% of_nat_diff
tff(fact_4555_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ) ).

% of_nat_diff
tff(fact_4556_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = minus_8373710615458151222nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ) ) ).

% of_nat_diff
tff(fact_4557_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M),aa_nat_complex(semiri8010041392384452111omplex,Na)) ) ) ).

% of_nat_diff
tff(fact_4558_ex__le__of__int,axiom,
    ! [Xa: real] :
    ? [Z3: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,Z3)) ).

% ex_le_of_int
tff(fact_4559_ex__le__of__int,axiom,
    ! [Xa: rat] :
    ? [Z3: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z3)) ).

% ex_le_of_int
tff(fact_4560_ex__less__of__int,axiom,
    ! [Xa: real] :
    ? [Z3: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,Z3)) ).

% ex_less_of_int
tff(fact_4561_ex__less__of__int,axiom,
    ! [Xa: rat] :
    ? [Z3: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z3)) ).

% ex_less_of_int
tff(fact_4562_ex__of__int__less,axiom,
    ! [Xa: real] :
    ? [Z3: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z3)),Xa) ).

% ex_of_int_less
tff(fact_4563_ex__of__int__less,axiom,
    ! [Xa: rat] :
    ? [Z3: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),Xa) ).

% ex_of_int_less
tff(fact_4564_mult__of__int__commute,axiom,
    ! [Xa: int,Y: real] : aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Xa)),Y) = aa_real_real(times_times_real(Y),aa_int_real(ring_1_of_int_real,Xa)) ).

% mult_of_int_commute
tff(fact_4565_mult__of__int__commute,axiom,
    ! [Xa: int,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,Xa)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_int_rat(ring_1_of_int_rat,Xa)) ).

% mult_of_int_commute
tff(fact_4566_mult__of__int__commute,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,ring_1_of_int_int(Xa)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),ring_1_of_int_int(Xa)) ).

% mult_of_int_commute
tff(fact_4567_powr__powr,axiom,
    ! [Xa: real,A: real,B: real] : aa_real_real(powr_real(aa_real_real(powr_real(Xa),A)),B) = aa_real_real(powr_real(Xa),aa_real_real(times_times_real(A),B)) ).

% powr_powr
tff(fact_4568_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( minus_minus_complex(A,B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_4569_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(minus_minus_real(A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_4570_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_4571_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_4572_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_mono
tff(fact_4573_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D)) ) ) ).

% diff_mono
tff(fact_4574_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_4575_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_left_mono
tff(fact_4576_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)) ) ).

% diff_left_mono
tff(fact_4577_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_4578_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_right_mono
tff(fact_4579_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) ) ).

% diff_right_mono
tff(fact_4580_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_4581_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4582_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4583_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4584_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_strict_mono
tff(fact_4585_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D)) ) ) ).

% diff_strict_mono
tff(fact_4586_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_4587_diff__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,D),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,C)),minus_8373710615458151222nteger(B,D)) ) ) ).

% diff_strict_mono
tff(fact_4588_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4589_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4590_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4591_diff__eq__diff__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( minus_8373710615458151222nteger(A,B) = minus_8373710615458151222nteger(C,D) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4592_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_strict_left_mono
tff(fact_4593_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)) ) ).

% diff_strict_left_mono
tff(fact_4594_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_4595_diff__strict__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(C,A)),minus_8373710615458151222nteger(C,B)) ) ).

% diff_strict_left_mono
tff(fact_4596_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_strict_right_mono
tff(fact_4597_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) ) ).

% diff_strict_right_mono
tff(fact_4598_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_4599_diff__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,C)),minus_8373710615458151222nteger(B,C)) ) ).

% diff_strict_right_mono
tff(fact_4600_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(minus_minus_real(A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4601_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4602_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4603_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_4604_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_4605_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_4606_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(minus_minus_real(C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4607_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4608_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4609_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_4610_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ).

% add_diff_eq
tff(fact_4611_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_4612_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_4613_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_diff_eq2
tff(fact_4614_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_4615_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_4616_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_add_eq
tff(fact_4617_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_4618_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4619_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4620_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4621_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(minus_minus_real(A),B) ) ) ).

% add_implies_diff
tff(fact_4622_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
     => ( C = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ) ) ).

% add_implies_diff
tff(fact_4623_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_4624_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_4625_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_4626_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_4627_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_4628_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_4629_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A),B)),aa_real_real(minus_minus_real(C),D)) ).

% add_diff_add
tff(fact_4630_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D)) ).

% add_diff_add
tff(fact_4631_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).

% add_diff_add
tff(fact_4632_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% left_diff_distrib
tff(fact_4633_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% left_diff_distrib
tff(fact_4634_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_4635_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib
tff(fact_4636_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib
tff(fact_4637_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_4638_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),C)),A) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(B),A)),aa_real_real(times_times_real(C),A)) ).

% left_diff_distrib'
tff(fact_4639_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)) ).

% left_diff_distrib'
tff(fact_4640_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_4641_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_4642_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib'
tff(fact_4643_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib'
tff(fact_4644_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_4645_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_4646_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D3: real,Q: fun_real_o] :
      ( ! [X: real,K2: real] :
          ( aa_real_o(P,X)
        <=> aa_real_o(P,aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K2),D3))) )
     => ( ! [X: real,K2: real] :
            ( aa_real_o(Q,X)
          <=> aa_real_o(Q,aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K2),D3))) )
       => ! [X5: real,K5: real] :
            ( ( aa_real_o(P,X5)
              | aa_real_o(Q,X5) )
          <=> ( aa_real_o(P,aa_real_real(minus_minus_real(X5),aa_real_real(times_times_real(K5),D3)))
              | aa_real_o(Q,aa_real_real(minus_minus_real(X5),aa_real_real(times_times_real(K5),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4647_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
      ( ! [X: rat,K2: rat] :
          ( aa_rat_o(P,X)
        <=> aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
     => ( ! [X: rat,K2: rat] :
            ( aa_rat_o(Q,X)
          <=> aa_rat_o(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
       => ! [X5: rat,K5: rat] :
            ( ( aa_rat_o(P,X5)
              | aa_rat_o(Q,X5) )
          <=> ( aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K5),D3)))
              | aa_rat_o(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K5),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4648_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X: int,K2: int] :
          ( aa_int_o(P,X)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
     => ( ! [X: int,K2: int] :
            ( aa_int_o(Q,X)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
       => ! [X5: int,K5: int] :
            ( ( aa_int_o(P,X5)
              | aa_int_o(Q,X5) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D3)))
              | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4649_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D3: real,Q: fun_real_o] :
      ( ! [X: real,K2: real] :
          ( aa_real_o(P,X)
        <=> aa_real_o(P,aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K2),D3))) )
     => ( ! [X: real,K2: real] :
            ( aa_real_o(Q,X)
          <=> aa_real_o(Q,aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K2),D3))) )
       => ! [X5: real,K5: real] :
            ( ( aa_real_o(P,X5)
              & aa_real_o(Q,X5) )
          <=> ( aa_real_o(P,aa_real_real(minus_minus_real(X5),aa_real_real(times_times_real(K5),D3)))
              & aa_real_o(Q,aa_real_real(minus_minus_real(X5),aa_real_real(times_times_real(K5),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4650_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
      ( ! [X: rat,K2: rat] :
          ( aa_rat_o(P,X)
        <=> aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
     => ( ! [X: rat,K2: rat] :
            ( aa_rat_o(Q,X)
          <=> aa_rat_o(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
       => ! [X5: rat,K5: rat] :
            ( ( aa_rat_o(P,X5)
              & aa_rat_o(Q,X5) )
          <=> ( aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K5),D3)))
              & aa_rat_o(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K5),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4651_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X: int,K2: int] :
          ( aa_int_o(P,X)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
     => ( ! [X: int,K2: int] :
            ( aa_int_o(Q,X)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
       => ! [X5: int,K5: int] :
            ( ( aa_int_o(P,X5)
              & aa_int_o(Q,X5) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D3)))
              & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4652_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : divide1717551699836669952omplex(minus_minus_complex(A,B),C) = minus_minus_complex(divide1717551699836669952omplex(A,C),divide1717551699836669952omplex(B,C)) ).

% diff_divide_distrib
tff(fact_4653_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ).

% diff_divide_distrib
tff(fact_4654_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) ).

% diff_divide_distrib
tff(fact_4655_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_right_eq
tff(fact_4656_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_right_eq
tff(fact_4657_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_left_eq
tff(fact_4658_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_left_eq
tff(fact_4659_mod__diff__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),B4)),C) ) ) ) ).

% mod_diff_cong
tff(fact_4660_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B4,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A3,B4),C) ) ) ) ).

% mod_diff_cong
tff(fact_4661_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_eq
tff(fact_4662_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_eq
tff(fact_4663_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_4664_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M) = zero_zero_nat )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_4665_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_4666_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: 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),Na)),K) ) ).

% less_imp_diff_less
tff(fact_4667_diff__diff__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na) ) ) ).

% diff_diff_less
tff(fact_4668_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ).

% Nat.diff_cancel
tff(fact_4669_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ).

% diff_cancel2
tff(fact_4670_diff__add__inverse,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_4671_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_4672_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ).

% diff_le_mono2
tff(fact_4673_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_4674_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),M) ).

% diff_le_self
tff(fact_4675_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),L)) ) ).

% diff_le_mono
tff(fact_4676_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_4677_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% le_diff_iff
tff(fact_4678_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_4679_diff__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% diff_mult_distrib
tff(fact_4680_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% diff_mult_distrib2
tff(fact_4681_of__int__ceiling__diff__one__le,axiom,
    ! [R: 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,R))),one_one_real)),R) ).

% of_int_ceiling_diff_one_le
tff(fact_4682_of__int__ceiling__diff__one__le,axiom,
    ! [R: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R))),one_one_rat)),R) ).

% of_int_ceiling_diff_one_le
tff(fact_4683_ceiling__correct,axiom,
    ! [Xa: 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,Xa))),one_one_real)),Xa)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xa))) ) ).

% ceiling_correct
tff(fact_4684_ceiling__correct,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xa))),one_one_rat)),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xa))) ) ).

% ceiling_correct
tff(fact_4685_ceiling__unique,axiom,
    ! [Z: int,Xa: 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)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,Xa) = Z ) ) ) ).

% ceiling_unique
tff(fact_4686_ceiling__unique,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xa) = Z ) ) ) ).

% ceiling_unique
tff(fact_4687_ceiling__eq__iff,axiom,
    ! [Xa: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xa) = A )
    <=> ( 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,A)),one_one_real)),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_4688_ceiling__eq__iff,axiom,
    ! [Xa: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,Xa) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_4689_ceiling__split,axiom,
    ! [P: fun_int_o,T: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,T))
    <=> ! [I2: 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,I2)),one_one_real)),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_4690_ceiling__split,axiom,
    ! [P: fun_int_o,T: rat] :
      ( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,T))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,I2)),one_one_rat)),T)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_4691_ceiling__less__iff,axiom,
    ! [Xa: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_4692_ceiling__less__iff,axiom,
    ! [Xa: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_4693_le__ceiling__iff,axiom,
    ! [Z: int,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> 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)),Xa) ) ).

% le_ceiling_iff
tff(fact_4694_le__ceiling__iff,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xa) ) ).

% le_ceiling_iff
tff(fact_4695_powr__non__neg,axiom,
    ! [A: real,Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(A),Xa)),zero_zero_real) ).

% powr_non_neg
tff(fact_4696_powr__less__mono2__neg,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Y),A)),aa_real_real(powr_real(Xa),A)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_4697_powr__mono2,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_mono2
tff(fact_4698_powr__ge__pzero,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(powr_real(Xa),Y)) ).

% powr_ge_pzero
tff(fact_4699_powr__less__cancel,axiom,
    ! [Xa: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel
tff(fact_4700_powr__less__mono,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B)) ) ) ).

% powr_less_mono
tff(fact_4701_powr__mono,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B)) ) ) ).

% powr_mono
tff(fact_4702_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_4703_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_4704_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_4705_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_4706_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_4707_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_4708_less__iff__diff__less__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,B)),zero_z3403309356797280102nteger) ) ).

% less_iff_diff_less_0
tff(fact_4709_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,Na: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4710_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,Na: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4711_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,Na: 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)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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)),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4712_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,Na: 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)),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),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)),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),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,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4713_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(minus_minus_real(Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4714_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4715_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,Na: 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)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4716_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,Na: 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)),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4717_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4718_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_4719_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4720_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4721_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4722_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4723_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4724_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4725_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_4726_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_4727_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% le_diff_eq
tff(fact_4728_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% le_diff_eq
tff(fact_4729_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% le_diff_eq
tff(fact_4730_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_le_eq
tff(fact_4731_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_le_eq
tff(fact_4732_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_4733_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4734_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4735_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4736_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4737_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( plus_p5714425477246183910nteger(B,minus_8373710615458151222nteger(A,B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4738_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_less_eq
tff(fact_4739_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_less_eq
tff(fact_4740_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_4741_diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,B)),C)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),plus_p5714425477246183910nteger(C,B)) ) ).

% diff_less_eq
tff(fact_4742_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% less_diff_eq
tff(fact_4743_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% less_diff_eq
tff(fact_4744_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% less_diff_eq
tff(fact_4745_less__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),minus_8373710615458151222nteger(C,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),C) ) ).

% less_diff_eq
tff(fact_4746_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B),E)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4747_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4748_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4749_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B),E)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4750_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4751_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4752_square__diff__square__factored,axiom,
    ! [Xa: real,Y: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xa),Xa)),aa_real_real(times_times_real(Y),Y)) = aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_real_real(minus_minus_real(Xa),Y)) ).

% square_diff_square_factored
tff(fact_4753_square__diff__square__factored,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),Y)) ).

% square_diff_square_factored
tff(fact_4754_square__diff__square__factored,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)) ).

% square_diff_square_factored
tff(fact_4755_mult__diff__mult,axiom,
    ! [Xa: real,Y: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xa),Y)),aa_real_real(times_times_real(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xa),aa_real_real(minus_minus_real(Y),B))),aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xa),A)),B)) ).

% mult_diff_mult
tff(fact_4756_mult__diff__mult,axiom,
    ! [Xa: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),A)),B)) ).

% mult_diff_mult
tff(fact_4757_mult__diff__mult,axiom,
    ! [Xa: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),A)),B)) ).

% mult_diff_mult
tff(fact_4758_diff__shunt__var,axiom,
    ! [Xa: set_real,Y: set_real] :
      ( ( minus_minus_set_real(Xa,Y) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xa),Y) ) ).

% diff_shunt_var
tff(fact_4759_diff__shunt__var,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( minus_minus_set_int(Xa,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),Y) ) ).

% diff_shunt_var
tff(fact_4760_diff__shunt__var,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Xa),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) ) ).

% diff_shunt_var
tff(fact_4761_le__of__int__ceiling,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xa))) ).

% le_of_int_ceiling
tff(fact_4762_le__of__int__ceiling,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xa))) ).

% le_of_int_ceiling
tff(fact_4763_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),M) ) ) ).

% diff_less
tff(fact_4764_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_4765_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_4766_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_4767_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ) ).

% less_diff_iff
tff(fact_4768_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_4769_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_4770_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_4771_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_4772_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_4773_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_4774_nat__minus__mod,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),M))),M) = zero_zero_nat ).

% nat_minus_mod
tff(fact_4775_mod__nat__sub,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Y) ) ) ).

% mod_nat_sub
tff(fact_4776_mod__geq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) ) ) ).

% mod_geq
tff(fact_4777_mod__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ).

% mod_if
tff(fact_4778_nat__minus__mod__plus__right,axiom,
    ! [Na: nat,Xa: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),M))),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),M) ).

% nat_minus_mod_plus_right
tff(fact_4779_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_4780_ceiling__divide__lower,axiom,
    ! [Q2: real,P3: 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(P3),Q2)))),one_one_real)),Q2)),P3) ) ).

% ceiling_divide_lower
tff(fact_4781_ceiling__divide__lower,axiom,
    ! [Q2: rat,P3: 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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q2)))),one_one_rat)),Q2)),P3) ) ).

% ceiling_divide_lower
tff(fact_4782_powr__less__mono2,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_less_mono2
tff(fact_4783_powr__mono2_H,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Y),A)),aa_real_real(powr_real(Xa),A)) ) ) ) ).

% powr_mono2'
tff(fact_4784_powr__inj,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(powr_real(A),Xa) = aa_real_real(powr_real(A),Y) )
        <=> ( Xa = Y ) ) ) ) ).

% powr_inj
tff(fact_4785_gr__one__powr,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(powr_real(Xa),Y)) ) ) ).

% gr_one_powr
tff(fact_4786_powr__le1,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_4787_powr__mono__both,axiom,
    ! [A: real,B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_4788_ge__one__powr__ge__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(powr_real(Xa),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_4789_powr__divide,axiom,
    ! [Xa: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(powr_real(aa_real_real(divide_divide_real(Xa),Y)),A) = aa_real_real(divide_divide_real(aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_divide
tff(fact_4790_powr__mult,axiom,
    ! [Xa: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(powr_real(aa_real_real(times_times_real(Xa),Y)),A) = aa_real_real(times_times_real(aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_mult
tff(fact_4791_log__base__powr,axiom,
    ! [A: real,B: real,Xa: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(log(aa_real_real(powr_real(A),B)),Xa) = aa_real_real(divide_divide_real(aa_real_real(log(A),Xa)),B) ) ) ).

% log_base_powr
tff(fact_4792_log__powr,axiom,
    ! [Xa: real,B: real,Y: real] :
      ( ( Xa != zero_zero_real )
     => ( aa_real_real(log(B),aa_real_real(powr_real(Xa),Y)) = aa_real_real(times_times_real(Y),aa_real_real(log(B),Xa)) ) ) ).

% log_powr
tff(fact_4793_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),E)),C)),aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4794_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4795_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4796_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),E)),C)),aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4797_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4798_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4799_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),E)),C)),aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_4800_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_4801_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_4802_less__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(times_3573771949741848930nteger(A,E),C)),plus_p5714425477246183910nteger(times_3573771949741848930nteger(B,E),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B),E),C)),D) ) ).

% less_add_iff1
tff(fact_4803_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),E)),C)),aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_4804_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_4805_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_4806_less__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(times_3573771949741848930nteger(A,E),C)),plus_p5714425477246183910nteger(times_3573771949741848930nteger(B,E),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),plus_p5714425477246183910nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(B,A),E),D)) ) ).

% less_add_iff2
tff(fact_4807_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( minus_minus_complex(divide1717551699836669952omplex(Xa,Z),Y) = divide1717551699836669952omplex(minus_minus_complex(Xa,aa_complex_complex(times_times_complex(Y),Z)),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4808_divide__diff__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xa),Z)),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xa),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4809_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4810_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( minus_minus_complex(Xa,divide1717551699836669952omplex(Y,Z)) = divide1717551699836669952omplex(minus_minus_complex(aa_complex_complex(times_times_complex(Xa),Z),Y),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4811_diff__divide__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(Xa),aa_real_real(divide_divide_real(Y),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xa),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4812_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4813_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xa: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( minus_minus_complex(divide1717551699836669952omplex(Xa,Y),divide1717551699836669952omplex(W,Z)) = divide1717551699836669952omplex(minus_minus_complex(aa_complex_complex(times_times_complex(Xa),Z),aa_complex_complex(times_times_complex(W),Y)),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4814_diff__frac__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xa),Y)),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(Xa),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4815_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4816_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      minus_minus_complex(A,divide1717551699836669952omplex(B,Z)) = $ite(Z = zero_zero_complex,A,divide1717551699836669952omplex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z),B),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4817_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4818_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4819_square__diff__one__factored,axiom,
    ! [Xa: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xa),Xa)),one_one_real) = aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),one_one_real)),aa_real_real(minus_minus_real(Xa),one_one_real)) ).

% square_diff_one_factored
tff(fact_4820_square__diff__one__factored,axiom,
    ! [Xa: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),one_one_rat)) ).

% square_diff_one_factored
tff(fact_4821_square__diff__one__factored,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),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,Xa),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),one_one_int)) ).

% square_diff_one_factored
tff(fact_4822_ceiling__le,axiom,
    ! [Xa: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),A) ) ).

% ceiling_le
tff(fact_4823_ceiling__le,axiom,
    ! [Xa: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),A) ) ).

% ceiling_le
tff(fact_4824_ceiling__le__iff,axiom,
    ! [Xa: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_4825_ceiling__le__iff,axiom,
    ! [Xa: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_4826_zmde,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% zmde
tff(fact_4827_zmde,axiom,
    ! [B: code_integer,A: code_integer] : times_3573771949741848930nteger(B,divide6298287555418463151nteger(A,B)) = minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) ).

% zmde
tff(fact_4828_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4829_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4830_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,times_3573771949741848930nteger(B,divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4831_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4832_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4833_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = times_3573771949741848930nteger(B,divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4834_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4835_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4836_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = times_3573771949741848930nteger(divide6298287555418463151nteger(A,B),B) ).

% minus_mod_eq_div_mult
tff(fact_4837_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4838_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4839_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,times_3573771949741848930nteger(divide6298287555418463151nteger(A,B),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4840_less__ceiling__iff,axiom,
    ! [Z: int,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xa) ) ).

% less_ceiling_iff
tff(fact_4841_less__ceiling__iff,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xa) ) ).

% less_ceiling_iff
tff(fact_4842_powr__add,axiom,
    ! [Xa: real,A: real,B: real] : aa_real_real(powr_real(Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(times_times_real(aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B)) ).

% powr_add
tff(fact_4843_real__of__int__div4,axiom,
    ! [Na: int,Xa: 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,Na),Xa))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,Xa))) ).

% real_of_int_div4
tff(fact_4844_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D4: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
           => aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split
tff(fact_4845_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D4: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
              & ~ aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_4846_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_4847_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_4848_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4849_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_4850_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_4851_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_4852_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_4853_mod__nat__add,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y)),Z) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y)),Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_4854_modulo__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na)) ).

% modulo_nat_def
tff(fact_4855_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_4856_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_4857_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_4858_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),ring_1_of_int_int(Z)) ) ).

% of_int_nonneg
tff(fact_4859_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_4860_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_4861_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),ring_1_of_int_int(Z)) ) ).

% of_int_pos
tff(fact_4862_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ) ).

% of_int_pos
tff(fact_4863_powr__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na) ) ) ).

% powr_realpow
tff(fact_4864_less__log__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(log(B),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),Y)),Xa) ) ) ) ).

% less_log_iff
tff(fact_4865_log__less__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),Xa)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(powr_real(B),Y)) ) ) ) ).

% log_less_iff
tff(fact_4866_less__powr__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(powr_real(B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),Xa)),Y) ) ) ) ).

% less_powr_iff
tff(fact_4867_powr__less__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),Y)),Xa)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(log(B),Xa)) ) ) ) ).

% powr_less_iff
tff(fact_4868_floor__exists1,axiom,
    ! [Xa: real] :
    ? [X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),Xa)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X ) ) ) ).

% floor_exists1
tff(fact_4869_floor__exists1,axiom,
    ! [Xa: rat] :
    ? [X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),Xa)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X ) ) ) ).

% floor_exists1
tff(fact_4870_floor__exists,axiom,
    ! [Xa: real] :
    ? [Z3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z3)),Xa)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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_4871_floor__exists,axiom,
    ! [Xa: rat] :
    ? [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),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_4872_of__int__ceiling__le__add__one,axiom,
    ! [R: 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,R))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_4873_of__int__ceiling__le__add__one,axiom,
    ! [R: 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,R))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,R),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_4874_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_int_real(ring_1_of_int_real,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xa) ) ).

% of_nat_less_of_int_iff
tff(fact_4875_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),ring_1_of_int_int(Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xa) ) ).

% of_nat_less_of_int_iff
tff(fact_4876_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_int_rat(ring_1_of_int_rat,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xa) ) ).

% of_nat_less_of_int_iff
tff(fact_4877_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xa: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xa) ) ).

% of_nat_less_of_int_iff
tff(fact_4878_frac__le__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xa),Y)),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(Xa),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_4879_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_4880_frac__less__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xa),Y)),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(Xa),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_4881_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_4882_power2__commute,axiom,
    ! [Xa: complex,Y: complex] : aa_nat_complex(power_power_complex(minus_minus_complex(Xa,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(minus_minus_complex(Y,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_4883_power2__commute,axiom,
    ! [Xa: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(Xa,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(Y,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_4884_power2__commute,axiom,
    ! [Xa: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Y),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_4885_power2__commute,axiom,
    ! [Xa: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_4886_power2__commute,axiom,
    ! [Xa: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_4887_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real)) ) ).

% int_le_real_less
tff(fact_4888_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real)),aa_int_real(ring_1_of_int_real,M)) ) ).

% int_less_real_le
tff(fact_4889_power__diff,axiom,
    ! [A: code_integer,Na: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ) ) ).

% power_diff
tff(fact_4890_power__diff,axiom,
    ! [A: complex,Na: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M),aa_nat_complex(power_power_complex(A),Na)) ) ) ) ).

% power_diff
tff(fact_4891_power__diff,axiom,
    ! [A: real,Na: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_diff
tff(fact_4892_power__diff,axiom,
    ! [A: rat,Na: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_diff
tff(fact_4893_power__diff,axiom,
    ! [A: nat,Na: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_diff
tff(fact_4894_power__diff,axiom,
    ! [A: int,Na: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_diff
tff(fact_4895_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)))),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% field_char_0_class.of_nat_div
tff(fact_4896_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)))),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% field_char_0_class.of_nat_div
tff(fact_4897_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = divide1717551699836669952omplex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))),aa_nat_complex(semiri8010041392384452111omplex,Na)) ).

% field_char_0_class.of_nat_div
tff(fact_4898_real__of__int__div__aux,axiom,
    ! [Xa: int,D: int] : aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Xa)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),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,Xa),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_4899_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Na))) ).

% mult_eq_if
tff(fact_4900_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_4901_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_4902_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ) ) ).

% nat_mult_power_less_eq
tff(fact_4903_powr__mult__base,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(times_times_real(Xa),aa_real_real(powr_real(Xa),Y)) = aa_real_real(powr_real(Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_4904_le__log__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(log(B),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),Y)),Xa) ) ) ) ).

% le_log_iff
tff(fact_4905_log__le__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),Xa)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(powr_real(B),Y)) ) ) ) ).

% log_le_iff
tff(fact_4906_le__powr__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(powr_real(B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),Xa)),Y) ) ) ) ).

% le_powr_iff
tff(fact_4907_powr__le__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),Y)),Xa)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(log(B),Xa)) ) ) ) ).

% powr_le_iff
tff(fact_4908_scaling__mono,axiom,
    ! [U: real,V: real,R: real,Sa: 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),R)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),Sa)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(divide_divide_real(aa_real_real(times_times_real(R),aa_real_real(minus_minus_real(V),U))),Sa))),V) ) ) ) ).

% scaling_mono
tff(fact_4909_scaling__mono,axiom,
    ! [U: rat,V: rat,R: rat,Sa: 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),R)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R),Sa)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,V),U))),Sa))),V) ) ) ) ).

% scaling_mono
tff(fact_4910_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) != 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,Na),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4911_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4912_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) != zero_z3403309356797280102nteger )
     => ( 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,Na),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4913_power__diff__power__eq,axiom,
    ! [A: code_integer,M: nat,Na: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4914_power__diff__power__eq,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4915_power__diff__power__eq,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4916_power__eq__if,axiom,
    ! [P3: assn,M: nat] :
      aa_nat_assn(power_power_assn(P3),M) = $ite(M = zero_zero_nat,one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P3),aa_nat_assn(power_power_assn(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4917_power__eq__if,axiom,
    ! [P3: complex,M: nat] :
      aa_nat_complex(power_power_complex(P3),M) = $ite(M = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4918_power__eq__if,axiom,
    ! [P3: code_integer,M: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(P3),M) = $ite(M = zero_zero_nat,one_one_Code_integer,times_3573771949741848930nteger(P3,aa_nat_Code_integer(power_8256067586552552935nteger(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4919_power__eq__if,axiom,
    ! [P3: real,M: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(times_times_real(P3),aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4920_power__eq__if,axiom,
    ! [P3: rat,M: nat] :
      aa_nat_rat(power_power_rat(P3),M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4921_power__eq__if,axiom,
    ! [P3: nat,M: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4922_power__eq__if,axiom,
    ! [P3: int,M: nat] :
      aa_nat_int(power_power_int(P3),M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4923_power__minus__mult,axiom,
    ! [Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Na) ) ) ).

% power_minus_mult
tff(fact_4924_power__minus__mult,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),A) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) ) ) ).

% power_minus_mult
tff(fact_4925_power__minus__mult,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) ) ) ).

% power_minus_mult
tff(fact_4926_power__minus__mult,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Na) ) ) ).

% power_minus_mult
tff(fact_4927_power__minus__mult,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) ) ) ).

% power_minus_mult
tff(fact_4928_power__minus__mult,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_int(power_power_int(A),Na) ) ) ).

% power_minus_mult
tff(fact_4929_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Na))) ) ).

% diff_le_diff_pow
tff(fact_4930_power__sub,axiom,
    ! [Na: nat,M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_sub
tff(fact_4931_ceiling__divide__upper,axiom,
    ! [Q2: real,P3: 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,P3),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(P3),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_4932_ceiling__divide__upper,axiom,
    ! [Q2: rat,P3: 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,P3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_4933_add__log__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(log(B),Xa)) = aa_real_real(log(B),aa_real_real(times_times_real(aa_real_real(powr_real(B),Y)),Xa)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_4934_log__add__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(B),Xa)),Y) = aa_real_real(log(B),aa_real_real(times_times_real(Xa),aa_real_real(powr_real(B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_4935_nat__power__less__diff,axiom,
    ! [Na: nat,Q2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ) ).

% nat_power_less_diff
tff(fact_4936_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_4937_nat__le__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_le_power_trans
tff(fact_4938_power__mod__div,axiom,
    ! [Xa: nat,Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))) ).

% power_mod_div
tff(fact_4939_ceiling__eq,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,Xa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_4940_ceiling__eq,axiom,
    ! [Na: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Na)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Na)),one_one_rat))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_4941_power2__diff,axiom,
    ! [Xa: complex,Y: complex] : aa_nat_complex(power_power_complex(minus_minus_complex(Xa,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_diff
tff(fact_4942_power2__diff,axiom,
    ! [Xa: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_diff
tff(fact_4943_power2__diff,axiom,
    ! [Xa: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_diff
tff(fact_4944_power2__diff,axiom,
    ! [Xa: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_diff
tff(fact_4945_power2__diff,axiom,
    ! [Xa: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(Xa,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_3573771949741848930nteger(times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),Xa),Y)) ).

% power2_diff
tff(fact_4946_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4947_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4948_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( modulo364778990260209775nteger(times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = times_3573771949741848930nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4949_less__two__pow__divD,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),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))),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))) ) ) ).

% less_two_pow_divD
tff(fact_4950_less__two__pow__divI,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),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))),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ).

% less_two_pow_divI
tff(fact_4951_nat__less__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_less_power_trans
tff(fact_4952_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4953_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4954_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4955_small__powers__of__2,axiom,
    ! [Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),one_one_nat))) ) ).

% small_powers_of_2
tff(fact_4956_cnt__cnt__eq,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_cnt(T) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,T)) ).

% cnt_cnt_eq
tff(fact_4957_space__cnt,axiom,
    ! [T: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one)))),vEBT_VEBT_cnt(T))) ).

% space_cnt
tff(fact_4958_Tb__T__vebt__buildupi_H_H,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(vEBT_VEBT_Tb2,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi''
tff(fact_4959_t__build__cnt,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Na))),aa_real_real(times_times_real(vEBT_VEBT_cnt(vEBT_vebt_buildup(Na))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_build_cnt
tff(fact_4960_t__buildup__cnt,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Na))),aa_real_real(times_times_real(vEBT_VEBT_cnt(vEBT_vebt_buildup(Na))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_buildup_cnt
tff(fact_4961_T__vebt__buildupi__cnt_H,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V441764108873111860ildupi,Na))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),vEBT_VEBT_cnt(vEBT_vebt_buildup(Na)))) ).

% T_vebt_buildupi_cnt'
tff(fact_4962_count__buildup_H,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,vEBT_VEBT_cnt(vEBT_vebt_buildup(Na))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) ).

% count_buildup'
tff(fact_4963_count__buildup,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,vEBT_VEBT_cnt(vEBT_vebt_buildup(Na))),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_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na))) ).

% count_buildup
tff(fact_4964_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_4965_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_4966_Diff__cancel,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_4967_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_4968_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_4969_empty__Diff,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_4970_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_4971_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_4972_Diff__empty,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_4973_insert__Diff1,axiom,
    ! [Xa: real,B2: set_real,A2: set_real] :
      ( aa_set_real_o(member_real(Xa),B2)
     => ( minus_minus_set_real(insert_real(Xa,A2),B2) = minus_minus_set_real(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4974_insert__Diff1,axiom,
    ! [Xa: int,B2: set_int,A2: set_int] :
      ( aa_set_int_o(member_int(Xa),B2)
     => ( minus_minus_set_int(insert_int(Xa,A2),B2) = minus_minus_set_int(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4975_insert__Diff1,axiom,
    ! [Xa: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B2)
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xa,A2),B2) = minus_5127226145743854075T_VEBT(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4976_insert__Diff1,axiom,
    ! [Xa: nat,B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(Xa,A2)),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) ) ) ).

% insert_Diff1
tff(fact_4977_Diff__insert0,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( minus_minus_set_real(A2,insert_real(Xa,B2)) = minus_minus_set_real(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4978_Diff__insert0,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( minus_minus_set_int(A2,insert_int(Xa,B2)) = minus_minus_set_int(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4979_Diff__insert0,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2)
     => ( minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xa,B2)) = minus_5127226145743854075T_VEBT(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4980_Diff__insert0,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) ) ) ).

% Diff_insert0
tff(fact_4981_Un__Diff__cancel2,axiom,
    ! [B2: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2) ).

% Un_Diff_cancel2
tff(fact_4982_Un__Diff__cancel,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ).

% Un_Diff_cancel
tff(fact_4983_idiff__0__right,axiom,
    ! [Na: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(minus_3235023915231533773d_enat,Na),zero_z5237406670263579293d_enat) = Na ).

% idiff_0_right
tff(fact_4984_idiff__0,axiom,
    ! [Na: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(minus_3235023915231533773d_enat,zero_z5237406670263579293d_enat),Na) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_4985_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A2,B2) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_4986_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_4987_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_4988_insert__Diff__single,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] : insert_VEBT_VEBT(A,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = insert_VEBT_VEBT(A,A2) ).

% insert_Diff_single
tff(fact_4989_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = insert_real(A,A2) ).

% insert_Diff_single
tff(fact_4990_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ).

% insert_Diff_single
tff(fact_4991_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A2) ).

% insert_Diff_single
tff(fact_4992_Tb_H__cnt,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_VEBT_Tb2,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,vEBT_vebt_buildup(Na)))) ).

% Tb'_cnt
tff(fact_4993_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_4994_diff__nat__numeral,axiom,
    ! [V: num,V2: 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,V2)) = 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,V2))) ).

% diff_nat_numeral
tff(fact_4995_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_4996_Diff__mono,axiom,
    ! [A2: set_nat,C2: set_nat,D3: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D3),B2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),D3)) ) ) ).

% Diff_mono
tff(fact_4997_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),A2) ).

% Diff_subset
tff(fact_4998_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2)) = A2 ) ) ) ).

% double_diff
tff(fact_4999_insert__Diff__if,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      minus_minus_set_real(insert_real(Xa,A2),B2) = $ite(aa_set_real_o(member_real(Xa),B2),minus_minus_set_real(A2,B2),insert_real(Xa,minus_minus_set_real(A2,B2))) ).

% insert_Diff_if
tff(fact_5000_insert__Diff__if,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      minus_minus_set_int(insert_int(Xa,A2),B2) = $ite(aa_set_int_o(member_int(Xa),B2),minus_minus_set_int(A2,B2),insert_int(Xa,minus_minus_set_int(A2,B2))) ).

% insert_Diff_if
tff(fact_5001_insert__Diff__if,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xa,A2),B2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B2),minus_5127226145743854075T_VEBT(A2,B2),insert_VEBT_VEBT(Xa,minus_5127226145743854075T_VEBT(A2,B2))) ).

% insert_Diff_if
tff(fact_5002_insert__Diff__if,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(Xa,A2)),B2) = $ite(aa_set_nat_o(member_nat(Xa),B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2),insert_nat(Xa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))) ).

% insert_Diff_if
tff(fact_5003_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_5004_set__diff__diff__left,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ).

% set_diff_diff_left
tff(fact_5005_Un__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C2)) ).

% Un_Diff
tff(fact_5006_int__distrib_I3_J,axiom,
    ! [Z1: int,Z2: 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),Z2)),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,Z2),W)) ).

% int_distrib(3)
tff(fact_5007_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z2: 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),Z2)) = 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),Z2)) ).

% int_distrib(4)
tff(fact_5008_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M4)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_diff_cases
tff(fact_5009_Word_Omod__minus__cong,axiom,
    ! [B: int,B4: int,Xa: int,X6: int,Y: int,Y5: int,Z4: int] :
      ( ( B = B4 )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),B4) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X6),B4) )
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y),B4) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y5),B4) )
         => ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),Y5) = Z4 )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Z4),B4) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_5010_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B2)
     => ? [B5: real] : aa_set_real_o(member_real(B5),minus_minus_set_real(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5011_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
     => ? [B5: int] : aa_set_int_o(member_int(B5),minus_minus_set_int(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5012_psubset__imp__ex__mem,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B2)
     => ? [B5: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B5),minus_5127226145743854075T_VEBT(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5013_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ? [B5: nat] : aa_set_nat_o(member_nat(B5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5014_subset__minus__empty,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( minus_minus_set_real(A2,B2) = bot_bot_set_real ) ) ).

% subset_minus_empty
tff(fact_5015_subset__minus__empty,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( minus_minus_set_int(A2,B2) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_5016_subset__minus__empty,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_5017_set__minus__singleton__eq,axiom,
    ! [Xa: vEBT_VEBT,X7: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),X7)
     => ( minus_5127226145743854075T_VEBT(X7,insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT)) = X7 ) ) ).

% set_minus_singleton_eq
tff(fact_5018_set__minus__singleton__eq,axiom,
    ! [Xa: real,X7: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),X7)
     => ( minus_minus_set_real(X7,insert_real(Xa,bot_bot_set_real)) = X7 ) ) ).

% set_minus_singleton_eq
tff(fact_5019_set__minus__singleton__eq,axiom,
    ! [Xa: int,X7: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),X7)
     => ( minus_minus_set_int(X7,insert_int(Xa,bot_bot_set_int)) = X7 ) ) ).

% set_minus_singleton_eq
tff(fact_5020_set__minus__singleton__eq,axiom,
    ! [Xa: nat,X7: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),X7)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X7),insert_nat(Xa,bot_bot_set_nat)) = X7 ) ) ).

% set_minus_singleton_eq
tff(fact_5021_insert__minus__eq,axiom,
    ! [Xa: vEBT_VEBT,Y: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( Xa != Y )
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xa,A2),insert_VEBT_VEBT(Y,bot_bo8194388402131092736T_VEBT)) = insert_VEBT_VEBT(Xa,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Y,bot_bo8194388402131092736T_VEBT))) ) ) ).

% insert_minus_eq
tff(fact_5022_insert__minus__eq,axiom,
    ! [Xa: real,Y: real,A2: set_real] :
      ( ( Xa != Y )
     => ( minus_minus_set_real(insert_real(Xa,A2),insert_real(Y,bot_bot_set_real)) = insert_real(Xa,minus_minus_set_real(A2,insert_real(Y,bot_bot_set_real))) ) ) ).

% insert_minus_eq
tff(fact_5023_insert__minus__eq,axiom,
    ! [Xa: int,Y: int,A2: set_int] :
      ( ( Xa != Y )
     => ( minus_minus_set_int(insert_int(Xa,A2),insert_int(Y,bot_bot_set_int)) = insert_int(Xa,minus_minus_set_int(A2,insert_int(Y,bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_5024_insert__minus__eq,axiom,
    ! [Xa: nat,Y: nat,A2: set_nat] :
      ( ( Xa != Y )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(Xa,A2)),insert_nat(Y,bot_bot_set_nat)) = insert_nat(Xa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Y,bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_5025_Diff__insert__absorb,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2)
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xa,A2),insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5026_Diff__insert__absorb,axiom,
    ! [Xa: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( minus_minus_set_real(insert_real(Xa,A2),insert_real(Xa,bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5027_Diff__insert__absorb,axiom,
    ! [Xa: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( minus_minus_set_int(insert_int(Xa,A2),insert_int(Xa,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5028_Diff__insert__absorb,axiom,
    ! [Xa: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(Xa,A2)),insert_nat(Xa,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5029_Diff__insert2,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,B2)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)),B2) ).

% Diff_insert2
tff(fact_5030_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real(A,bot_bot_set_real)),B2) ).

% Diff_insert2
tff(fact_5031_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int(A,bot_bot_set_int)),B2) ).

% Diff_insert2
tff(fact_5032_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,bot_bot_set_nat))),B2) ).

% Diff_insert2
tff(fact_5033_insert__Diff,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
     => ( insert_VEBT_VEBT(A,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = A2 ) ) ).

% insert_Diff
tff(fact_5034_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_5035_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_5036_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5037_Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,B2)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A2,B2),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)) ).

% Diff_insert
tff(fact_5038_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,B2),insert_real(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_5039_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,B2),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_5040_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_5041_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,Xa: real,C2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B2,insert_real(Xa,C2)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B2,C2))
        & ~ aa_set_real_o(member_real(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_5042_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,Xa: int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B2,insert_int(Xa,C2)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B2,C2))
        & ~ aa_set_int_o(member_int(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_5043_subset__Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,Xa: vEBT_VEBT,C2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B2,insert_VEBT_VEBT(Xa,C2)))
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B2,C2))
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_5044_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,Xa: nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),insert_nat(Xa,C2)))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C2))
        & ~ aa_set_nat_o(member_nat(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_5045_Diff__partition,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = B2 ) ) ).

% Diff_partition
tff(fact_5046_Diff__subset__conv,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C2)
    <=> aa_set_nat_o(ord_less_eq_set_nat(A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ) ).

% Diff_subset_conv
tff(fact_5047_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_5048_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_5049_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xa: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(plus_p3455044024723400733d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(minus_3235023915231533773d_enat,Y),Z)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(minus_3235023915231533773d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(plus_p3455044024723400733d_enat,Xa),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_5050_Bolzano,axiom,
    ! [A: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A4: real,B5: real,C4: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A4),B5)
           => ( aa_real_o(aa_real_fun_real_o(P,B5),C4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),B5)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B5),C4)
                 => aa_real_o(aa_real_fun_real_o(P,A4),C4) ) ) ) )
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
               => ? [D5: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
                    & ! [A4: real,B5: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),X)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B5)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(B5),A4)),D5) )
                       => aa_real_o(aa_real_fun_real_o(P,A4),B5) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% Bolzano
tff(fact_5051_Diff__single__insert,axiom,
    ! [A2: set_VEBT_VEBT,Xa: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT))),B2)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(Xa,B2)) ) ).

% Diff_single_insert
tff(fact_5052_Diff__single__insert,axiom,
    ! [A2: set_real,Xa: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))),B2)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,B2)) ) ).

% Diff_single_insert
tff(fact_5053_Diff__single__insert,axiom,
    ! [A2: set_int,Xa: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(Xa,bot_bot_set_int))),B2)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,B2)) ) ).

% Diff_single_insert
tff(fact_5054_Diff__single__insert,axiom,
    ! [A2: set_nat,Xa: nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xa,B2)) ) ).

% Diff_single_insert
tff(fact_5055_subset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,Xa: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(Xa,B2))
    <=> $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT))),B2),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5056_subset__insert__iff,axiom,
    ! [A2: set_real,Xa: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,B2))
    <=> $ite(aa_set_real_o(member_real(Xa),A2),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5057_subset__insert__iff,axiom,
    ! [A2: set_int,Xa: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,B2))
    <=> $ite(aa_set_int_o(member_int(Xa),A2),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(Xa,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5058_subset__insert__iff,axiom,
    ! [A2: set_nat,Xa: nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xa,B2))
    <=> $ite(aa_set_nat_o(member_nat(Xa),A2),aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))),B2),aa_set_nat_o(ord_less_eq_set_nat(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5059_plusinfinity,axiom,
    ! [D: int,P2: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P2,X)
          <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z5: int] :
            ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X)
             => ( aa_int_o(P,X)
              <=> aa_int_o(P2,X) ) )
         => ( ? [X_13: int] : aa_int_o(P2,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_5060_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)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P1,X)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z5: int] :
            ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z5)
             => ( aa_int_o(P,X)
              <=> aa_int_o(P1,X) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_5061_remove__subset,axiom,
    ! [Xa: vEBT_VEBT,S: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),S)
     => aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT))),S) ) ).

% remove_subset
tff(fact_5062_remove__subset,axiom,
    ! [Xa: real,S: set_real] :
      ( aa_set_real_o(member_real(Xa),S)
     => aa_set_real_o(ord_less_set_real(minus_minus_set_real(S,insert_real(Xa,bot_bot_set_real))),S) ) ).

% remove_subset
tff(fact_5063_remove__subset,axiom,
    ! [Xa: int,S: set_int] :
      ( aa_set_int_o(member_int(Xa),S)
     => aa_set_int_o(ord_less_set_int(minus_minus_set_int(S,insert_int(Xa,bot_bot_set_int))),S) ) ).

% remove_subset
tff(fact_5064_remove__subset,axiom,
    ! [Xa: nat,S: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),S)
     => aa_set_nat_o(ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(Xa,bot_bot_set_nat))),S) ) ).

% remove_subset
tff(fact_5065_less__1__helper,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),one_one_int)),M) ) ).

% less_1_helper
tff(fact_5066_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
           => ( aa_int_o(P,I3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_5067_int__mod__le_H,axiom,
    ! [B: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Na))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Na)) ) ).

% int_mod_le'
tff(fact_5068_int__minus,axiom,
    ! [Na: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_5069_mod__div__equality__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_equality_div_eq
tff(fact_5070_nat__minus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_minus_as_int
tff(fact_5071_psubset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,Xa: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),insert_VEBT_VEBT(Xa,B2))
    <=> $ite(
          aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B2),
          aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B2),
          $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2),aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT))),B2),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5072_psubset__insert__iff,axiom,
    ! [A2: set_real,Xa: real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),insert_real(Xa,B2))
    <=> $ite(
          aa_set_real_o(member_real(Xa),B2),
          aa_set_real_o(ord_less_set_real(A2),B2),
          $ite(aa_set_real_o(member_real(Xa),A2),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5073_psubset__insert__iff,axiom,
    ! [A2: set_int,Xa: int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),insert_int(Xa,B2))
    <=> $ite(
          aa_set_int_o(member_int(Xa),B2),
          aa_set_int_o(ord_less_set_int(A2),B2),
          $ite(aa_set_int_o(member_int(Xa),A2),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A2,insert_int(Xa,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5074_psubset__insert__iff,axiom,
    ! [A2: set_nat,Xa: nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),insert_nat(Xa,B2))
    <=> $ite(
          aa_set_nat_o(member_nat(Xa),B2),
          aa_set_nat_o(ord_less_set_nat(A2),B2),
          $ite(aa_set_nat_o(member_nat(Xa),A2),aa_set_nat_o(ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))),B2),aa_set_nat_o(ord_less_eq_set_nat(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5075_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_5076_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)
     => ( ! [X: int] :
            ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X5: int] :
              ( aa_int_o(P,X5)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_5077_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z4),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_5078_nat__diff__distrib_H,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_5079_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_5080_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xa: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_5081_real__of__nat__div2,axiom,
    ! [Na: nat,Xa: 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,Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xa)))) ).

% real_of_nat_div2
tff(fact_5082_real__of__nat__div3,axiom,
    ! [Na: nat,Xa: 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,Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xa)))),one_one_real) ).

% real_of_nat_div3
tff(fact_5083_log__divide,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(log(A),aa_real_real(divide_divide_real(Xa),Y)) = aa_real_real(minus_minus_real(aa_real_real(log(A),Xa)),aa_real_real(log(A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_5084_mod__add__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_5085_mod__sub__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_5086_real__of__int__div2,axiom,
    ! [Na: int,Xa: 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,Na)),aa_int_real(ring_1_of_int_real,Xa))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xa)))) ).

% real_of_int_div2
tff(fact_5087_real__of__int__div3,axiom,
    ! [Na: int,Xa: 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,Na)),aa_int_real(ring_1_of_int_real,Xa))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xa)))),one_one_real) ).

% real_of_int_div3
tff(fact_5088_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4),
          $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_5089_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_5090_minus__log__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(minus_minus_real(Y),aa_real_real(log(B),Xa)) = aa_real_real(log(B),aa_real_real(divide_divide_real(aa_real_real(powr_real(B),Y)),Xa)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_5091_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_5092_space__space_H,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T)) ).

% space_space'
tff(fact_5093_Tb__T__vebt__buildupi,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Na))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi
tff(fact_5094_Tb__T__vebt__buildupi_H,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(vEBT_V9176841429113362141ildupi,Na)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi'
tff(fact_5095_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_5096_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_5097_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)),one_one_real))) ) ).

% space_2_pow_bound
tff(fact_5098_Tb__Tb_H,axiom,
    ! [T: nat] : aa_nat_int(vEBT_VEBT_Tb,T) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_VEBT_Tb2,T)) ).

% Tb_Tb'
tff(fact_5099_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(T),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_5100_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(T),zero_zero_nat) ).

% valid_0_not
tff(fact_5101_deg__not__0,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% deg_not_0
tff(fact_5102_buildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Na)),Na) ) ).

% buildup_gives_valid
tff(fact_5103_invar__vebt__buildup,axiom,
    ! [Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Na)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% invar_vebt_buildup
tff(fact_5104_DiffI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),A2)
     => ( ~ aa_set_real_o(member_real(C),B2)
       => aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2)) ) ) ).

% DiffI
tff(fact_5105_DiffI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => ( ~ aa_set_int_o(member_int(C),B2)
       => aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2)) ) ) ).

% DiffI
tff(fact_5106_DiffI,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B2)) ) ) ).

% DiffI
tff(fact_5107_DiffI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),A2)
     => ( ~ aa_set_nat_o(member_nat(C),B2)
       => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) ) ) ).

% DiffI
tff(fact_5108_Diff__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
    <=> ( aa_set_real_o(member_real(C),A2)
        & ~ aa_set_real_o(member_real(C),B2) ) ) ).

% Diff_iff
tff(fact_5109_Diff__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
    <=> ( aa_set_int_o(member_int(C),A2)
        & ~ aa_set_int_o(member_int(C),B2) ) ) ).

% Diff_iff
tff(fact_5110_Diff__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B2))
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ) ).

% Diff_iff
tff(fact_5111_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))
    <=> ( aa_set_nat_o(member_nat(C),A2)
        & ~ aa_set_nat_o(member_nat(C),B2) ) ) ).

% Diff_iff
tff(fact_5112_Diff__idemp,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) ).

% Diff_idemp
tff(fact_5113_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(vEBT_invar_vebt(vEBT_VEBT_insert(T,Xa)),Na) ) ).

% insert'_pres_valid
tff(fact_5114_space_H__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space'_bound
tff(fact_5115_space__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space_bound
tff(fact_5116_vebt__space__linear__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% vebt_space_linear_bound
tff(fact_5117_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,vEBT_VEBT_cnt(T)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)),one_one_real))) ) ).

% cnt_bound'
tff(fact_5118_cnt__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,vEBT_VEBT_cnt(T)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)),aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))))) ) ).

% cnt_bound
tff(fact_5119_set__diff__eq,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_co(A2),B2)) ).

% set_diff_eq
tff(fact_5120_set__diff__eq,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B2) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_cp(A2),B2)) ).

% set_diff_eq
tff(fact_5121_set__diff__eq,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_cq(A2),B2)) ).

% set_diff_eq
tff(fact_5122_set__diff__eq,axiom,
    ! [A2: set_complex,B2: set_complex] : minus_811609699411566653omplex(A2,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_cr(A2),B2)) ).

% set_diff_eq
tff(fact_5123_set__diff__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A2,B2) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_cs(A2),B2)) ).

% set_diff_eq
tff(fact_5124_set__diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ct(A2),B2)) ).

% set_diff_eq
tff(fact_5125_DiffE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
     => ~ ( aa_set_real_o(member_real(C),A2)
         => aa_set_real_o(member_real(C),B2) ) ) ).

% DiffE
tff(fact_5126_DiffE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
     => ~ ( aa_set_int_o(member_int(C),A2)
         => aa_set_int_o(member_int(C),B2) ) ) ).

% DiffE
tff(fact_5127_DiffE,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B2))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ) ).

% DiffE
tff(fact_5128_DiffE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))
     => ~ ( aa_set_nat_o(member_nat(C),A2)
         => aa_set_nat_o(member_nat(C),B2) ) ) ).

% DiffE
tff(fact_5129_DiffD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
     => aa_set_real_o(member_real(C),A2) ) ).

% DiffD1
tff(fact_5130_DiffD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
     => aa_set_int_o(member_int(C),A2) ) ).

% DiffD1
tff(fact_5131_DiffD1,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B2))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2) ) ).

% DiffD1
tff(fact_5132_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))
     => aa_set_nat_o(member_nat(C),A2) ) ).

% DiffD1
tff(fact_5133_DiffD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
     => ~ aa_set_real_o(member_real(C),B2) ) ).

% DiffD2
tff(fact_5134_DiffD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
     => ~ aa_set_int_o(member_int(C),B2) ) ).

% DiffD2
tff(fact_5135_DiffD2,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),minus_5127226145743854075T_VEBT(A2,B2))
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ).

% DiffD2
tff(fact_5136_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2))
     => ~ aa_set_nat_o(member_nat(C),B2) ) ).

% DiffD2
tff(fact_5137_minus__set__def,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(minus_404528632214381484real_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).

% minus_set_def
tff(fact_5138_minus__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B2) = collect_VEBT_VEBT(minus_4582106842932979072VEBT_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% minus_set_def
tff(fact_5139_minus__set__def,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).

% minus_set_def
tff(fact_5140_minus__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] : minus_811609699411566653omplex(A2,B2) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2))) ).

% minus_set_def
tff(fact_5141_minus__set__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A2,B2) = collec213857154873943460nt_int(minus_827889751674334651_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% minus_set_def
tff(fact_5142_minus__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% minus_set_def
tff(fact_5143_mult__commute__abs,axiom,
    ! [C: real] : aTP_Lamm_cu(C) = times_times_real(C) ).

% mult_commute_abs
tff(fact_5144_mult__commute__abs,axiom,
    ! [C: rat] : aa_rat_fun_rat_rat(aTP_Lamm_cv,C) = aa_rat_fun_rat_rat(times_times_rat,C) ).

% mult_commute_abs
tff(fact_5145_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_cw,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_5146_mult__commute__abs,axiom,
    ! [C: int] : aa_int_fun_int_int(aTP_Lamm_cx,C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_5147_time__vebt__succ,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => 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(T),Xa))),aa_real_real(aa_rea7340605660300375871l_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))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)))))) ) ).

% time_vebt_succ
tff(fact_5148_time__vebt__pred,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => 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(T),Xa))),aa_real_real(aa_rea7340605660300375871l_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))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)))))) ) ).

% time_vebt_pred
tff(fact_5149_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))),aa_real_real(times_times_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(bit0,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))))) ) ) ).

% delete_bound_size_univ
tff(fact_5150_time__vebt__delete,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))),aa_real_real(times_times_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(bit0,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))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)))))) ) ).

% time_vebt_delete
tff(fact_5151_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(T),Xa))),aa_real_real(aa_rea7340605660300375871l_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)))) ) ) ).

% delete_bound_size_univ'
tff(fact_5152_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: 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))),Deg) )
     => ( aa_nat_o(vEBT_invar_vebt(T),Deg)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_real_real(aa_rea7340605660300375871l_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_5153_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)) = 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,Na))) ) ) ).

% heigt_uplog_rel
tff(fact_5154_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% delete_bound_height'
tff(fact_5155_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => 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,T))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ).

% two_powr_height_bound_deg
tff(fact_5156_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),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,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ) ).

% delete_bound_height
tff(fact_5157_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => 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(T),Xa))),aa_real_real(aa_rea7340605660300375871l_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_5158_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => 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(T),Xa))),aa_real_real(aa_rea7340605660300375871l_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_5159_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c(T),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,T))),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_5160_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d(T),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,T))),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_5161_valid__pres__insert,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_insert(T,Xa)),Na) ) ) ).

% valid_pres_insert
tff(fact_5162_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xa: nat,Na: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xa)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% member_bound
tff(fact_5163_artanh__def,axiom,
    ! [Xa: real] : aa_real_real(artanh_real,Xa) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(minus_minus_real(one_one_real),Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_5164_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
         => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(T,Xa)),Y)
           => ( aa_nat_o(vEBT_vebt_member(T),Y)
              | ( Xa = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_5165_ln__inj__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( ( aa_real_real(ln_ln_real,Xa) = aa_real_real(ln_ln_real,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_5166_ln__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_5167_ln__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_5168_ln__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_5169_ln__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% ln_gt_zero_iff
tff(fact_5170_ln__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( ( aa_real_real(ln_ln_real,Xa) = zero_zero_real )
      <=> ( Xa = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_5171_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_5172_ln__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_5173_ln__ge__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% ln_ge_zero_iff
tff(fact_5174_ln__less__self,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),Xa) ) ).

% ln_less_self
tff(fact_5175_log__def,axiom,
    ! [A: real,Xa: real] : aa_real_real(log(A),Xa) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,A)) ).

% log_def
tff(fact_5176_ln__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),Xa) ) ).

% ln_bound
tff(fact_5177_ln__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa)) ) ).

% ln_gt_zero
tff(fact_5178_ln__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_5179_ln__gt__zero__imp__gt__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_5180_ln__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa)) ) ).

% ln_ge_zero
tff(fact_5181_ln__powr,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(powr_real(Xa),Y)) = aa_real_real(times_times_real(Y),aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_powr
tff(fact_5182_ln__ge__zero__imp__ge__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_5183_ln__add__one__self__le__self,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa) ) ).

% ln_add_one_self_le_self
tff(fact_5184_ln__mult,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(times_times_real(Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_5185_ln__eq__minus__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( ( aa_real_real(ln_ln_real,Xa) = aa_real_real(minus_minus_real(Xa),one_one_real) )
       => ( Xa = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_5186_ln__div,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(Xa),Y)) = aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_5187_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_5188_ln__le__minus__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(minus_minus_real(Xa),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_5189_ln__diff__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xa),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_5190_ln__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_realpow
tff(fact_5191_ln__powr__bound,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(divide_divide_real(aa_real_real(powr_real(Xa),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_5192_ln__powr__bound2,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(aa_real_real(ln_ln_real,Xa)),A)),aa_real_real(times_times_real(aa_real_real(powr_real(A),A)),Xa)) ) ) ).

% ln_powr_bound2
tff(fact_5193_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
             => ( aa_real_real(log(A),Xa) = 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,A))),aa_real_real(log(B),Xa)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_5194_ln__one__plus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_5195_insert__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_VEBT_set_vebt(T)),insert_nat(Xa,bot_bot_set_nat)) = vEBT_VEBT_set_vebt(vEBT_vebt_insert(T,Xa)) ) ) ) ).

% insert_corr
tff(fact_5196_insert__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_set_vebt(T)),insert_nat(Xa,bot_bot_set_nat)) = vEBT_set_vebt(vEBT_vebt_insert(T,Xa)) ) ) ) ).

% insert_correct
tff(fact_5197_set__vebt__insert,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( vEBT_set_vebt(vEBT_vebt_insert(T,Xa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_set_vebt(T)),insert_nat(Xa,bot_bot_set_nat)) ) ) ) ).

% set_vebt_insert
tff(fact_5198_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_inserti(Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_insert(T,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,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_5199_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_inserti(Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_insert(T,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,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% htt_vebt_inserti
tff(fact_5200_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => 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(T),Xa))),aa_real_real(aa_rea7340605660300375871l_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_5201_VEBT__internal_OTb_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,zero_zero_nat) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_5202_set__vebt__equal,axiom,
    ! [T_1: vEBT_VEBT,Na: nat,T_2: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(T_1),Na)
     => ( aa_nat_o(vEBT_invar_vebt(T_2),Na)
       => ( ( T_1 = T_2 )
        <=> ( vEBT_set_vebt(T_1) = vEBT_set_vebt(T_2) ) ) ) ) ).

% set_vebt_equal
tff(fact_5203_unique__tree,axiom,
    ! [T: vEBT_VEBT,Na: nat,Sa: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_invar_vebt(Sa),Na)
       => ( ( vEBT_set_vebt(T) = vEBT_set_vebt(Sa) )
         => ( Sa = T ) ) ) ) ).

% unique_tree
tff(fact_5204_uniquetree,axiom,
    ! [T: vEBT_VEBT,Na: nat,Sa: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_invar_vebt(Sa),Na)
       => ( ( vEBT_VEBT_set_vebt(T) = vEBT_VEBT_set_vebt(Sa) )
         => ( Sa = T ) ) ) ) ).

% uniquetree
tff(fact_5205_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_5206_set__vebt__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_vebt_member(T),Xa)
      <=> aa_set_nat_o(member_nat(Xa),vEBT_set_vebt(T)) ) ) ).

% set_vebt_member
tff(fact_5207_set__vebt__buildup,axiom,
    ! [I: nat] : vEBT_set_vebt(vEBT_vebt_buildup(I)) = bot_bot_set_nat ).

% set_vebt_buildup
tff(fact_5208_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_5209_buildup__gives__empty,axiom,
    ! [Na: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Na)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_5210_succ__member,axiom,
    ! [T: vEBT_VEBT,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),Xa),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(T),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        & ! [Z6: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(T),Z6)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z6) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z6) ) ) ) ).

% succ_member
tff(fact_5211_pred__member,axiom,
    ! [T: vEBT_VEBT,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),Xa),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(T),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
        & ! [Z6: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(T),Z6)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z6),Xa) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z6),Y) ) ) ) ).

% pred_member
tff(fact_5212_member__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r(T),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,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% member_bound_height
tff(fact_5213_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_5214_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi'_univ
tff(fact_5215_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi_univ
tff(fact_5216_htt__vebt__buildupi_H,axiom,
    ! [Na: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% htt_vebt_buildupi'
tff(fact_5217_htt__vebt__buildupi,axiom,
    ! [Na: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% htt_vebt_buildupi
tff(fact_5218_inrange,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(T)),aa_nat_set_nat(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))),Na)),one_one_nat))) ) ).

% inrange
tff(fact_5219_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => 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(T),Xa))),aa_real_real(aa_rea7340605660300375871l_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_5220_htt__vebt__deletei,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_deletei(Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_delete(T,Xa))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_int_nat(nat2,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,Na))))))) ) ).

% htt_vebt_deletei
tff(fact_5221_invar__vebt__delete,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(T,Xa)),Na) ) ).

% invar_vebt_delete
tff(fact_5222_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_delete(T,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa_nat_o(vEBT_vebt_member(T),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_5223_set__vebt__delete,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( vEBT_set_vebt(vEBT_vebt_delete(T,Xa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,vEBT_set_vebt(T)),insert_nat(Xa,bot_bot_set_nat)) ) ) ).

% set_vebt_delete
tff(fact_5224_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,Xa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,vEBT_VEBT_set_vebt(T)),insert_nat(Xa,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_5225_delete__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,Xa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,vEBT_set_vebt(T)),insert_nat(Xa,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_5226_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),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,T))),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_5227_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( vEBT_set_vebt(vEBT_VEBT_insert(T,Xa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_set_vebt(T)),insert_nat(Xa,bot_bot_set_nat))),aa_nat_set_nat(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))),Na)),one_one_nat))) ) ) ).

% insert'_correct
tff(fact_5228_T__vebt__buildupi,axiom,
    ! [Na: nat,H: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(vEBT_V739175172307565963ildupi(Na),H)),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% T_vebt_buildupi
tff(fact_5229_atLeastAtMost__singleton,axiom,
    ! [A: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(A),A) = insert_nat(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_5230_atLeastAtMost__singleton,axiom,
    ! [A: int] : set_or1266510415728281911st_int(A,A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_5231_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_5232_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = insert_nat(C,bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5233_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( set_or1266510415728281911st_int(A,B) = insert_int(C,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5234_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( set_or1222579329274155063t_real(A,B) = insert_real(C,bot_bot_set_real) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5235_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_5236_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_5237_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( set_or189985376899183464nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
tff(fact_5238_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_5239_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_5240_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_5241_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_5242_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_5243_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_5244_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_5245_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_5246_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_5247_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_5248_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_5249_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_5250_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_5251_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_5252_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_5253_inf__right__idem,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) ).

% inf_right_idem
tff(fact_5254_inf__right__idem,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) ).

% inf_right_idem
tff(fact_5255_inf__right__idem,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y) ).

% inf_right_idem
tff(fact_5256_inf__right__idem,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),Y) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y) ).

% inf_right_idem
tff(fact_5257_inf__right__idem,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),Y) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y) ).

% inf_right_idem
tff(fact_5258_inf_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).

% inf.right_idem
tff(fact_5259_inf_Oright__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).

% inf.right_idem
tff(fact_5260_inf_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ).

% inf.right_idem
tff(fact_5261_inf_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ).

% inf.right_idem
tff(fact_5262_inf_Oright__idem,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) ).

% inf.right_idem
tff(fact_5263_inf__left__idem,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) ).

% inf_left_idem
tff(fact_5264_inf__left__idem,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) ).

% inf_left_idem
tff(fact_5265_inf__left__idem,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y) ).

% inf_left_idem
tff(fact_5266_inf__left__idem,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y) ).

% inf_left_idem
tff(fact_5267_inf__left__idem,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y) ).

% inf_left_idem
tff(fact_5268_inf_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).

% inf.left_idem
tff(fact_5269_inf_Oleft__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).

% inf.left_idem
tff(fact_5270_inf_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ).

% inf.left_idem
tff(fact_5271_inf_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ).

% inf.left_idem
tff(fact_5272_inf_Oleft__idem,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) ).

% inf.left_idem
tff(fact_5273_inf__idem,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Xa) = Xa ).

% inf_idem
tff(fact_5274_inf__idem,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Xa) = Xa ).

% inf_idem
tff(fact_5275_inf__idem,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Xa) = Xa ).

% inf_idem
tff(fact_5276_inf__idem,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Xa) = Xa ).

% inf_idem
tff(fact_5277_inf__idem,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Xa) = Xa ).

% inf_idem
tff(fact_5278_inf_Oidem,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),A) = A ).

% inf.idem
tff(fact_5279_inf_Oidem,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),A) = A ).

% inf.idem
tff(fact_5280_inf_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),A) = A ).

% inf.idem
tff(fact_5281_inf_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,A),A) = A ).

% inf.idem
tff(fact_5282_inf_Oidem,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),A) = A ).

% inf.idem
tff(fact_5283_Int__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2))
    <=> ( aa_set_real_o(member_real(C),A2)
        & aa_set_real_o(member_real(C),B2) ) ) ).

% Int_iff
tff(fact_5284_Int__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2))
    <=> ( aa_set_int_o(member_int(C),A2)
        & aa_set_int_o(member_int(C),B2) ) ) ).

% Int_iff
tff(fact_5285_Int__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),inf_in473752961466201218T_VEBT(A2,B2))
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
        & aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ) ).

% Int_iff
tff(fact_5286_Int__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
    <=> ( aa_set_nat_o(member_nat(C),A2)
        & aa_set_nat_o(member_nat(C),B2) ) ) ).

% Int_iff
tff(fact_5287_IntI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),A2)
     => ( aa_set_real_o(member_real(C),B2)
       => aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2)) ) ) ).

% IntI
tff(fact_5288_IntI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => ( aa_set_int_o(member_int(C),B2)
       => aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2)) ) ) ).

% IntI
tff(fact_5289_IntI,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),inf_in473752961466201218T_VEBT(A2,B2)) ) ) ).

% IntI
tff(fact_5290_IntI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),A2)
     => ( aa_set_nat_o(member_nat(C),B2)
       => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% IntI
tff(fact_5291_inf_Obounded__iff,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ord_less_eq_assn(A,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C))
    <=> ( ord_less_eq_assn(A,B)
        & ord_less_eq_assn(A,C) ) ) ).

% inf.bounded_iff
tff(fact_5292_inf_Obounded__iff,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,B),C))
    <=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
        & aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),C) ) ) ).

% inf.bounded_iff
tff(fact_5293_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% inf.bounded_iff
tff(fact_5294_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_5295_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_5296_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% inf.bounded_iff
tff(fact_5297_le__inf__iff,axiom,
    ! [Xa: assn,Y: assn,Z: assn] :
      ( ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))
    <=> ( ord_less_eq_assn(Xa,Y)
        & ord_less_eq_assn(Xa,Z) ) ) ).

% le_inf_iff
tff(fact_5298_le__inf__iff,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z))
    <=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),Y)
        & aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),Z) ) ) ).

% le_inf_iff
tff(fact_5299_le__inf__iff,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
        & aa_set_nat_o(ord_less_eq_set_nat(Xa),Z) ) ) ).

% le_inf_iff
tff(fact_5300_le__inf__iff,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z) ) ) ).

% le_inf_iff
tff(fact_5301_le__inf__iff,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z) ) ) ).

% le_inf_iff
tff(fact_5302_le__inf__iff,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z) ) ) ).

% le_inf_iff
tff(fact_5303_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),bot_bot_assn) = bot_bot_assn ).

% boolean_algebra.conj_zero_right
tff(fact_5304_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_real] : inf_inf_set_real(Xa,bot_bot_set_real) = bot_bot_set_real ).

% boolean_algebra.conj_zero_right
tff(fact_5305_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),bot_bot_set_nat) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_right
tff(fact_5306_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_int] : inf_inf_set_int(Xa,bot_bot_set_int) = bot_bot_set_int ).

% boolean_algebra.conj_zero_right
tff(fact_5307_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),Xa) = bot_bot_assn ).

% boolean_algebra.conj_zero_left
tff(fact_5308_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_real] : inf_inf_set_real(bot_bot_set_real,Xa) = bot_bot_set_real ).

% boolean_algebra.conj_zero_left
tff(fact_5309_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),Xa) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_left
tff(fact_5310_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_int] : inf_inf_set_int(bot_bot_set_int,Xa) = bot_bot_set_int ).

% boolean_algebra.conj_zero_left
tff(fact_5311_inf__bot__right,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),bot_bot_assn) = bot_bot_assn ).

% inf_bot_right
tff(fact_5312_inf__bot__right,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),bot_bo4199563552545308370d_enat) = bot_bo4199563552545308370d_enat ).

% inf_bot_right
tff(fact_5313_inf__bot__right,axiom,
    ! [Xa: filter_nat] : inf_inf_filter_nat(Xa,bot_bot_filter_nat) = bot_bot_filter_nat ).

% inf_bot_right
tff(fact_5314_inf__bot__right,axiom,
    ! [Xa: set_real] : inf_inf_set_real(Xa,bot_bot_set_real) = bot_bot_set_real ).

% inf_bot_right
tff(fact_5315_inf__bot__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),bot_bot_set_nat) = bot_bot_set_nat ).

% inf_bot_right
tff(fact_5316_inf__bot__right,axiom,
    ! [Xa: set_int] : inf_inf_set_int(Xa,bot_bot_set_int) = bot_bot_set_int ).

% inf_bot_right
tff(fact_5317_inf__bot__left,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),Xa) = bot_bot_assn ).

% inf_bot_left
tff(fact_5318_inf__bot__left,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,bot_bo4199563552545308370d_enat),Xa) = bot_bo4199563552545308370d_enat ).

% inf_bot_left
tff(fact_5319_inf__bot__left,axiom,
    ! [Xa: filter_nat] : inf_inf_filter_nat(bot_bot_filter_nat,Xa) = bot_bot_filter_nat ).

% inf_bot_left
tff(fact_5320_inf__bot__left,axiom,
    ! [Xa: set_real] : inf_inf_set_real(bot_bot_set_real,Xa) = bot_bot_set_real ).

% inf_bot_left
tff(fact_5321_inf__bot__left,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),Xa) = bot_bot_set_nat ).

% inf_bot_left
tff(fact_5322_inf__bot__left,axiom,
    ! [Xa: set_int] : inf_inf_set_int(bot_bot_set_int,Xa) = bot_bot_set_int ).

% inf_bot_left
tff(fact_5323_sup__inf__absorb,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)) = Xa ).

% sup_inf_absorb
tff(fact_5324_sup__inf__absorb,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)) = Xa ).

% sup_inf_absorb
tff(fact_5325_sup__inf__absorb,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)) = Xa ).

% sup_inf_absorb
tff(fact_5326_sup__inf__absorb,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)) = Xa ).

% sup_inf_absorb
tff(fact_5327_sup__inf__absorb,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)) = Xa ).

% sup_inf_absorb
tff(fact_5328_inf__sup__absorb,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)) = Xa ).

% inf_sup_absorb
tff(fact_5329_inf__sup__absorb,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)) = Xa ).

% inf_sup_absorb
tff(fact_5330_inf__sup__absorb,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)) = Xa ).

% inf_sup_absorb
tff(fact_5331_inf__sup__absorb,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)) = Xa ).

% inf_sup_absorb
tff(fact_5332_inf__sup__absorb,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)) = Xa ).

% inf_sup_absorb
tff(fact_5333_Int__subset__iff,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(C2),A2)
        & aa_set_nat_o(ord_less_eq_set_nat(C2),B2) ) ) ).

% Int_subset_iff
tff(fact_5334_Int__insert__right__if1,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( inf_inf_set_real(A2,insert_real(A,B2)) = insert_real(A,inf_inf_set_real(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_5335_Int__insert__right__if1,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( inf_inf_set_int(A2,insert_int(A,B2)) = insert_int(A,inf_inf_set_int(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_5336_Int__insert__right__if1,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
     => ( inf_in473752961466201218T_VEBT(A2,insert_VEBT_VEBT(A,B2)) = insert_VEBT_VEBT(A,inf_in473752961466201218T_VEBT(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_5337_Int__insert__right__if1,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat(A,B2)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_5338_Int__insert__right__if0,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(A),A2)
     => ( inf_inf_set_real(A2,insert_real(A,B2)) = inf_inf_set_real(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_5339_Int__insert__right__if0,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(A),A2)
     => ( inf_inf_set_int(A2,insert_int(A,B2)) = inf_inf_set_int(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_5340_Int__insert__right__if0,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
     => ( inf_in473752961466201218T_VEBT(A2,insert_VEBT_VEBT(A,B2)) = inf_in473752961466201218T_VEBT(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_5341_Int__insert__right__if0,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_5342_insert__inter__insert,axiom,
    ! [A: int,A2: set_int,B2: set_int] : inf_inf_set_int(insert_int(A,A2),insert_int(A,B2)) = insert_int(A,inf_inf_set_int(A2,B2)) ).

% insert_inter_insert
tff(fact_5343_insert__inter__insert,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : inf_in473752961466201218T_VEBT(insert_VEBT_VEBT(A,A2),insert_VEBT_VEBT(A,B2)) = insert_VEBT_VEBT(A,inf_in473752961466201218T_VEBT(A2,B2)) ).

% insert_inter_insert
tff(fact_5344_insert__inter__insert,axiom,
    ! [A: real,A2: set_real,B2: set_real] : inf_inf_set_real(insert_real(A,A2),insert_real(A,B2)) = insert_real(A,inf_inf_set_real(A2,B2)) ).

% insert_inter_insert
tff(fact_5345_insert__inter__insert,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A2)),insert_nat(A,B2)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ).

% insert_inter_insert
tff(fact_5346_Int__insert__left__if1,axiom,
    ! [A: real,C2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(A),C2)
     => ( inf_inf_set_real(insert_real(A,B2),C2) = insert_real(A,inf_inf_set_real(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_5347_Int__insert__left__if1,axiom,
    ! [A: int,C2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(A),C2)
     => ( inf_inf_set_int(insert_int(A,B2),C2) = insert_int(A,inf_inf_set_int(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_5348_Int__insert__left__if1,axiom,
    ! [A: vEBT_VEBT,C2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),C2)
     => ( inf_in473752961466201218T_VEBT(insert_VEBT_VEBT(A,B2),C2) = insert_VEBT_VEBT(A,inf_in473752961466201218T_VEBT(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_5349_Int__insert__left__if1,axiom,
    ! [A: nat,C2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(A),C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B2)),C2) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_5350_Int__insert__left__if0,axiom,
    ! [A: real,C2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(A),C2)
     => ( inf_inf_set_real(insert_real(A,B2),C2) = inf_inf_set_real(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_5351_Int__insert__left__if0,axiom,
    ! [A: int,C2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(A),C2)
     => ( inf_inf_set_int(insert_int(A,B2),C2) = inf_inf_set_int(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_5352_Int__insert__left__if0,axiom,
    ! [A: vEBT_VEBT,C2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),C2)
     => ( inf_in473752961466201218T_VEBT(insert_VEBT_VEBT(A,B2),C2) = inf_in473752961466201218T_VEBT(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_5353_Int__insert__left__if0,axiom,
    ! [A: nat,C2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_5354_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T3)),S) = S ).

% Un_Int_eq(1)
tff(fact_5355_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T3)),T3) = T3 ).

% Un_Int_eq(2)
tff(fact_5356_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T3)) = S ).

% Un_Int_eq(3)
tff(fact_5357_Un__Int__eq_I4_J,axiom,
    ! [T3: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,T3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T3)) = T3 ).

% Un_Int_eq(4)
tff(fact_5358_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T3)),S) = S ).

% Int_Un_eq(1)
tff(fact_5359_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T3)),T3) = T3 ).

% Int_Un_eq(2)
tff(fact_5360_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T3)) = S ).

% Int_Un_eq(3)
tff(fact_5361_Int__Un__eq_I4_J,axiom,
    ! [T3: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,T3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T3)) = T3 ).

% Int_Un_eq(4)
tff(fact_5362_insert__disjoint_I1_J,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( inf_in473752961466201218T_VEBT(insert_VEBT_VEBT(A,A2),B2) = bot_bo8194388402131092736T_VEBT )
    <=> ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B2)
        & ( inf_in473752961466201218T_VEBT(A2,B2) = bot_bo8194388402131092736T_VEBT ) ) ) ).

% insert_disjoint(1)
tff(fact_5363_insert__disjoint_I1_J,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(insert_real(A,A2),B2) = bot_bot_set_real )
    <=> ( ~ aa_set_real_o(member_real(A),B2)
        & ( inf_inf_set_real(A2,B2) = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
tff(fact_5364_insert__disjoint_I1_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A2)),B2) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B2)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_5365_insert__disjoint_I1_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(insert_int(A,A2),B2) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B2)
        & ( inf_inf_set_int(A2,B2) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_5366_insert__disjoint_I2_J,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( bot_bo8194388402131092736T_VEBT = inf_in473752961466201218T_VEBT(insert_VEBT_VEBT(A,A2),B2) )
    <=> ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B2)
        & ( bot_bo8194388402131092736T_VEBT = inf_in473752961466201218T_VEBT(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_5367_insert__disjoint_I2_J,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(insert_real(A,A2),B2) )
    <=> ( ~ aa_set_real_o(member_real(A),B2)
        & ( bot_bot_set_real = inf_inf_set_real(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_5368_insert__disjoint_I2_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A2)),B2) )
    <=> ( ~ aa_set_nat_o(member_nat(A),B2)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_5369_insert__disjoint_I2_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(insert_int(A,A2),B2) )
    <=> ( ~ aa_set_int_o(member_int(A),B2)
        & ( bot_bot_set_int = inf_inf_set_int(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_5370_disjoint__insert_I1_J,axiom,
    ! [B2: set_VEBT_VEBT,A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( inf_in473752961466201218T_VEBT(B2,insert_VEBT_VEBT(A,A2)) = bot_bo8194388402131092736T_VEBT )
    <=> ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B2)
        & ( inf_in473752961466201218T_VEBT(B2,A2) = bot_bo8194388402131092736T_VEBT ) ) ) ).

% disjoint_insert(1)
tff(fact_5371_disjoint__insert_I1_J,axiom,
    ! [B2: set_real,A: real,A2: set_real] :
      ( ( inf_inf_set_real(B2,insert_real(A,A2)) = bot_bot_set_real )
    <=> ( ~ aa_set_real_o(member_real(A),B2)
        & ( inf_inf_set_real(B2,A2) = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
tff(fact_5372_disjoint__insert_I1_J,axiom,
    ! [B2: set_nat,A: nat,A2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),insert_nat(A,A2)) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B2)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_5373_disjoint__insert_I1_J,axiom,
    ! [B2: set_int,A: int,A2: set_int] :
      ( ( inf_inf_set_int(B2,insert_int(A,A2)) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B2)
        & ( inf_inf_set_int(B2,A2) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_5374_disjoint__insert_I2_J,axiom,
    ! [A2: set_VEBT_VEBT,B: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( bot_bo8194388402131092736T_VEBT = inf_in473752961466201218T_VEBT(A2,insert_VEBT_VEBT(B,B2)) )
    <=> ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),A2)
        & ( bot_bo8194388402131092736T_VEBT = inf_in473752961466201218T_VEBT(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_5375_disjoint__insert_I2_J,axiom,
    ! [A2: set_real,B: real,B2: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(A2,insert_real(B,B2)) )
    <=> ( ~ aa_set_real_o(member_real(B),A2)
        & ( bot_bot_set_real = inf_inf_set_real(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_5376_disjoint__insert_I2_J,axiom,
    ! [A2: set_nat,B: nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat(B,B2)) )
    <=> ( ~ aa_set_nat_o(member_nat(B),A2)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_5377_disjoint__insert_I2_J,axiom,
    ! [A2: set_int,B: int,B2: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(A2,insert_int(B,B2)) )
    <=> ( ~ aa_set_int_o(member_int(B),A2)
        & ( bot_bot_set_int = inf_inf_set_int(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_5378_Diff__disjoint,axiom,
    ! [A2: set_real,B2: set_real] : inf_inf_set_real(A2,minus_minus_set_real(B2,A2)) = bot_bot_set_real ).

% Diff_disjoint
tff(fact_5379_Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] : inf_inf_set_int(A2,minus_minus_set_int(B2,A2)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_5380_Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_5381_Int__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)) ).

% Int_left_commute
tff(fact_5382_Int__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% Int_left_absorb
tff(fact_5383_Int__commute,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2) ).

% Int_commute
tff(fact_5384_Int__absorb,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),A2) = A2 ).

% Int_absorb
tff(fact_5385_Int__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ).

% Int_assoc
tff(fact_5386_IntD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2))
     => aa_set_real_o(member_real(C),B2) ) ).

% IntD2
tff(fact_5387_IntD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2))
     => aa_set_int_o(member_int(C),B2) ) ).

% IntD2
tff(fact_5388_IntD2,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),inf_in473752961466201218T_VEBT(A2,B2))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ).

% IntD2
tff(fact_5389_IntD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => aa_set_nat_o(member_nat(C),B2) ) ).

% IntD2
tff(fact_5390_IntD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2))
     => aa_set_real_o(member_real(C),A2) ) ).

% IntD1
tff(fact_5391_IntD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2))
     => aa_set_int_o(member_int(C),A2) ) ).

% IntD1
tff(fact_5392_IntD1,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),inf_in473752961466201218T_VEBT(A2,B2))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2) ) ).

% IntD1
tff(fact_5393_IntD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => aa_set_nat_o(member_nat(C),A2) ) ).

% IntD1
tff(fact_5394_IntE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2))
     => ~ ( aa_set_real_o(member_real(C),A2)
         => ~ aa_set_real_o(member_real(C),B2) ) ) ).

% IntE
tff(fact_5395_IntE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2))
     => ~ ( aa_set_int_o(member_int(C),A2)
         => ~ aa_set_int_o(member_int(C),B2) ) ) ).

% IntE
tff(fact_5396_IntE,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),inf_in473752961466201218T_VEBT(A2,B2))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
         => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B2) ) ) ).

% IntE
tff(fact_5397_IntE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => ~ ( aa_set_nat_o(member_nat(C),A2)
         => ~ aa_set_nat_o(member_nat(C),B2) ) ) ).

% IntE
tff(fact_5398_inf__left__commute,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z)) ).

% inf_left_commute
tff(fact_5399_inf__left__commute,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Z)) ).

% inf_left_commute
tff(fact_5400_inf__left__commute,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z)) ).

% inf_left_commute
tff(fact_5401_inf__left__commute,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Z)) ).

% inf_left_commute
tff(fact_5402_inf__left__commute,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Z)) ).

% inf_left_commute
tff(fact_5403_inf_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ).

% inf.left_commute
tff(fact_5404_inf_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C)) ).

% inf.left_commute
tff(fact_5405_inf_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ).

% inf.left_commute
tff(fact_5406_inf_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,B),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),C)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C)) ).

% inf.left_commute
tff(fact_5407_inf_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),C)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,B),C)) ).

% inf.left_commute
tff(fact_5408_inf__commute,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Xa) ).

% inf_commute
tff(fact_5409_inf__commute,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Xa) ).

% inf_commute
tff(fact_5410_inf__commute,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Xa) ).

% inf_commute
tff(fact_5411_inf__commute,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Xa) ).

% inf_commute
tff(fact_5412_inf__commute,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Xa) ).

% inf_commute
tff(fact_5413_inf_Ocommute,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),A) ).

% inf.commute
tff(fact_5414_inf_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),A) ).

% inf.commute
tff(fact_5415_inf_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),A) ).

% inf.commute
tff(fact_5416_inf_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = aa_int_int(aa_int_fun_int_int(inf_inf_int,B),A) ).

% inf.commute
tff(fact_5417_inf_Ocommute,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,B),A) ).

% inf.commute
tff(fact_5418_inf__assoc,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ).

% inf_assoc
tff(fact_5419_inf__assoc,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_assoc
tff(fact_5420_inf__assoc,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ).

% inf_assoc
tff(fact_5421_inf__assoc,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),Z) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) ).

% inf_assoc
tff(fact_5422_inf__assoc,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),Z) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z)) ).

% inf_assoc
tff(fact_5423_inf_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ).

% inf.assoc
tff(fact_5424_inf_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C)) ).

% inf.assoc
tff(fact_5425_inf_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ).

% inf.assoc
tff(fact_5426_inf_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C)) ).

% inf.assoc
tff(fact_5427_inf_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),C) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,B),C)) ).

% inf.assoc
tff(fact_5428_inf__sup__aci_I1_J,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Xa) ).

% inf_sup_aci(1)
tff(fact_5429_inf__sup__aci_I1_J,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Xa) ).

% inf_sup_aci(1)
tff(fact_5430_inf__sup__aci_I1_J,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Xa) ).

% inf_sup_aci(1)
tff(fact_5431_inf__sup__aci_I1_J,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Xa) ).

% inf_sup_aci(1)
tff(fact_5432_inf__sup__aci_I1_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Xa) ).

% inf_sup_aci(1)
tff(fact_5433_inf__sup__aci_I2_J,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_5434_inf__sup__aci_I2_J,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_5435_inf__sup__aci_I2_J,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_5436_inf__sup__aci_I2_J,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),Z) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_5437_inf__sup__aci_I2_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),Z) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_5438_inf__sup__aci_I3_J,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z)) ).

% inf_sup_aci(3)
tff(fact_5439_inf__sup__aci_I3_J,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Z)) ).

% inf_sup_aci(3)
tff(fact_5440_inf__sup__aci_I3_J,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z)) ).

% inf_sup_aci(3)
tff(fact_5441_inf__sup__aci_I3_J,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Z)) ).

% inf_sup_aci(3)
tff(fact_5442_inf__sup__aci_I3_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Z)) ).

% inf_sup_aci(3)
tff(fact_5443_inf__sup__aci_I4_J,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) ).

% inf_sup_aci(4)
tff(fact_5444_inf__sup__aci_I4_J,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) ).

% inf_sup_aci(4)
tff(fact_5445_inf__sup__aci_I4_J,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y) ).

% inf_sup_aci(4)
tff(fact_5446_inf__sup__aci_I4_J,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y) ).

% inf_sup_aci(4)
tff(fact_5447_inf__sup__aci_I4_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y) ).

% inf_sup_aci(4)
tff(fact_5448_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_5449_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: assn,K: assn,B: assn,A: assn] :
      ( ( B2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),B) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_5450_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_5451_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) = aa_int_int(aa_int_fun_int_int(inf_inf_int,K),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_5452_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: extended_enat,K: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( B2 = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,K),B) )
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B2) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,K),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_5453_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),A) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_5454_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: assn,K: assn,A: assn,B: assn] :
      ( ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),A) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_5455_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_5456_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B) = aa_int_int(aa_int_fun_int_int(inf_inf_int,K),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_5457_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: extended_enat,K: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( A2 = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,K),A) )
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A2),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,K),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_5458_Collect__conj__eq,axiom,
    ! [P: fun_int_o,Q: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cy(P),Q)) = inf_inf_set_int(collect_int(P),collect_int(Q)) ).

% Collect_conj_eq
tff(fact_5459_Collect__conj__eq,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] : collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_cz(P),Q)) = inf_inf_set_complex(collect_complex(P),collect_complex(Q)) ).

% Collect_conj_eq
tff(fact_5460_Collect__conj__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_da(P),Q)) = inf_in2269163501485487111nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q)) ).

% Collect_conj_eq
tff(fact_5461_Collect__conj__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_db(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,collect_nat(P)),collect_nat(Q)) ).

% Collect_conj_eq
tff(fact_5462_Int__Collect,axiom,
    ! [Xa: real,A2: set_real,P: fun_real_o] :
      ( aa_set_real_o(member_real(Xa),inf_inf_set_real(A2,collect_real(P)))
    <=> ( aa_set_real_o(member_real(Xa),A2)
        & aa_real_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_5463_Int__Collect,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),inf_in473752961466201218T_VEBT(A2,collect_VEBT_VEBT(P)))
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A2)
        & aa_VEBT_VEBT_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_5464_Int__Collect,axiom,
    ! [Xa: int,A2: set_int,P: fun_int_o] :
      ( aa_set_int_o(member_int(Xa),inf_inf_set_int(A2,collect_int(P)))
    <=> ( aa_set_int_o(member_int(Xa),A2)
        & aa_int_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_5465_Int__Collect,axiom,
    ! [Xa: complex,A2: set_complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(Xa),inf_inf_set_complex(A2,collect_complex(P)))
    <=> ( aa_set_complex_o(member_complex(Xa),A2)
        & aa_complex_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_5466_Int__Collect,axiom,
    ! [Xa: product_prod_int_int,A2: set_Pr958786334691620121nt_int,P: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(Xa,inf_in2269163501485487111nt_int(A2,collec213857154873943460nt_int(P)))
    <=> ( member5262025264175285858nt_int(Xa,A2)
        & aa_Pro4592068669816069595_int_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_5467_Int__Collect,axiom,
    ! [Xa: nat,A2: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P)))
    <=> ( aa_set_nat_o(member_nat(Xa),A2)
        & aa_nat_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_5468_Int__def,axiom,
    ! [A2: set_real,B2: set_real] : inf_inf_set_real(A2,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_dc(A2),B2)) ).

% Int_def
tff(fact_5469_Int__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : inf_in473752961466201218T_VEBT(A2,B2) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_dd(A2),B2)) ).

% Int_def
tff(fact_5470_Int__def,axiom,
    ! [A2: set_int,B2: set_int] : inf_inf_set_int(A2,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_de(A2),B2)) ).

% Int_def
tff(fact_5471_Int__def,axiom,
    ! [A2: set_complex,B2: set_complex] : inf_inf_set_complex(A2,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_df(A2),B2)) ).

% Int_def
tff(fact_5472_Int__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : inf_in2269163501485487111nt_int(A2,B2) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_dg(A2),B2)) ).

% Int_def
tff(fact_5473_Int__def,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_dh(A2),B2)) ).

% Int_def
tff(fact_5474_inf_OcoboundedI2,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ord_less_eq_assn(B,C)
     => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),C) ) ).

% inf.coboundedI2
tff(fact_5475_inf_OcoboundedI2,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),C)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_5476_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_5477_inf_OcoboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_5478_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_5479_inf_OcoboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_5480_inf_OcoboundedI1,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ord_less_eq_assn(A,C)
     => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),C) ) ).

% inf.coboundedI1
tff(fact_5481_inf_OcoboundedI1,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),C)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_5482_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),C)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_5483_inf_OcoboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_5484_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_5485_inf_OcoboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_5486_inf_Oabsorb__iff2,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_5487_inf_Oabsorb__iff2,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_5488_inf_Oabsorb__iff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_5489_inf_Oabsorb__iff2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_5490_inf_Oabsorb__iff2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_5491_inf_Oabsorb__iff2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_5492_inf_Oabsorb__iff1,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_5493_inf_Oabsorb__iff1,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_5494_inf_Oabsorb__iff1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_5495_inf_Oabsorb__iff1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_5496_inf_Oabsorb__iff1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_5497_inf_Oabsorb__iff1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_5498_inf_Ocobounded2,axiom,
    ! [A: assn,B: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),B) ).

% inf.cobounded2
tff(fact_5499_inf_Ocobounded2,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),B) ).

% inf.cobounded2
tff(fact_5500_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_5501_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),B) ).

% inf.cobounded2
tff(fact_5502_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_5503_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),B) ).

% inf.cobounded2
tff(fact_5504_inf_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),A) ).

% inf.cobounded1
tff(fact_5505_inf_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),A) ).

% inf.cobounded1
tff(fact_5506_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_5507_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),A) ).

% inf.cobounded1
tff(fact_5508_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_5509_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),A) ).

% inf.cobounded1
tff(fact_5510_inf_Oorder__iff,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.order_iff
tff(fact_5511_inf_Oorder__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
    <=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) ) ) ).

% inf.order_iff
tff(fact_5512_inf_Oorder__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_5513_inf_Oorder__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.order_iff
tff(fact_5514_inf_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_5515_inf_Oorder__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.order_iff
tff(fact_5516_inf__greatest,axiom,
    ! [Xa: assn,Y: assn,Z: assn] :
      ( ord_less_eq_assn(Xa,Y)
     => ( ord_less_eq_assn(Xa,Z)
       => ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ) ) ).

% inf_greatest
tff(fact_5517_inf__greatest,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),Y)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),Z)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_5518_inf__greatest,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Z)
       => aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_5519_inf__greatest,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_5520_inf__greatest,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_5521_inf__greatest,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) ) ) ).

% inf_greatest
tff(fact_5522_inf_OboundedI,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ord_less_eq_assn(A,B)
     => ( ord_less_eq_assn(A,C)
       => ord_less_eq_assn(A,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C)) ) ) ).

% inf.boundedI
tff(fact_5523_inf_OboundedI,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),C)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,B),C)) ) ) ).

% inf.boundedI
tff(fact_5524_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),C)
       => aa_set_nat_o(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_5525_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C)) ) ) ).

% inf.boundedI
tff(fact_5526_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_5527_inf_OboundedI,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C)) ) ) ).

% inf.boundedI
tff(fact_5528_inf_OboundedE,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ord_less_eq_assn(A,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C))
     => ~ ( ord_less_eq_assn(A,B)
         => ~ ord_less_eq_assn(A,C) ) ) ).

% inf.boundedE
tff(fact_5529_inf_OboundedE,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,B),C))
     => ~ ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
         => ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),C) ) ) ).

% inf.boundedE
tff(fact_5530_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% inf.boundedE
tff(fact_5531_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% inf.boundedE
tff(fact_5532_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% inf.boundedE
tff(fact_5533_inf_OboundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% inf.boundedE
tff(fact_5534_inf__absorb2,axiom,
    ! [Y: assn,Xa: assn] :
      ( ord_less_eq_assn(Y,Xa)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) = Y ) ) ).

% inf_absorb2
tff(fact_5535_inf__absorb2,axiom,
    ! [Y: extended_enat,Xa: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y),Xa)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y) = Y ) ) ).

% inf_absorb2
tff(fact_5536_inf__absorb2,axiom,
    ! [Y: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),Xa)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) = Y ) ) ).

% inf_absorb2
tff(fact_5537_inf__absorb2,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Y) = Y ) ) ).

% inf_absorb2
tff(fact_5538_inf__absorb2,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y) = Y ) ) ).

% inf_absorb2
tff(fact_5539_inf__absorb2,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y) = Y ) ) ).

% inf_absorb2
tff(fact_5540_inf__absorb1,axiom,
    ! [Xa: assn,Y: assn] :
      ( ord_less_eq_assn(Xa,Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) = Xa ) ) ).

% inf_absorb1
tff(fact_5541_inf__absorb1,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),Y)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y) = Xa ) ) ).

% inf_absorb1
tff(fact_5542_inf__absorb1,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) = Xa ) ) ).

% inf_absorb1
tff(fact_5543_inf__absorb1,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Y) = Xa ) ) ).

% inf_absorb1
tff(fact_5544_inf__absorb1,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y) = Xa ) ) ).

% inf_absorb1
tff(fact_5545_inf__absorb1,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y) = Xa ) ) ).

% inf_absorb1
tff(fact_5546_inf_Oabsorb2,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb2
tff(fact_5547_inf_Oabsorb2,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_5548_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_5549_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_5550_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_5551_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb2
tff(fact_5552_inf_Oabsorb1,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb1
tff(fact_5553_inf_Oabsorb1,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_5554_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_5555_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_5556_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_5557_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb1
tff(fact_5558_le__iff__inf,axiom,
    ! [Xa: assn,Y: assn] :
      ( ord_less_eq_assn(Xa,Y)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) = Xa ) ) ).

% le_iff_inf
tff(fact_5559_le__iff__inf,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),Y)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y) = Xa ) ) ).

% le_iff_inf
tff(fact_5560_le__iff__inf,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) = Xa ) ) ).

% le_iff_inf
tff(fact_5561_le__iff__inf,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Y) = Xa ) ) ).

% le_iff_inf
tff(fact_5562_le__iff__inf,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y) = Xa ) ) ).

% le_iff_inf
tff(fact_5563_le__iff__inf,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y) = Xa ) ) ).

% le_iff_inf
tff(fact_5564_inf__unique,axiom,
    ! [F: fun_as3983628580203722482n_assn,Xa: assn,Y: assn] :
      ( ! [X: assn,Y2: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y2),X)
     => ( ! [X: assn,Y2: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y2),Y2)
       => ( ! [X: assn,Y2: assn,Z3: assn] :
              ( ord_less_eq_assn(X,Y2)
             => ( ord_less_eq_assn(X,Z3)
               => ord_less_eq_assn(X,aa_assn_assn(aa_ass954072293579886259n_assn(F,Y2),Z3)) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,Xa),Y) ) ) ) ) ).

% inf_unique
tff(fact_5565_inf__unique,axiom,
    ! [F: fun_Ex8341929632020906428d_enat,Xa: extended_enat,Y: extended_enat] :
      ( ! [X: extended_enat,Y2: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(F,X),Y2)),X)
     => ( ! [X: extended_enat,Y2: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(F,X),Y2)),Y2)
       => ( ! [X: extended_enat,Y2: extended_enat,Z3: extended_enat] :
              ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(X),Y2)
             => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(X),Z3)
               => aa_Extended_enat_o(ord_le2932123472753598470d_enat(X),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(F,Y2),Z3)) ) )
         => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(F,Xa),Y) ) ) ) ) ).

% inf_unique
tff(fact_5566_inf__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,Xa: set_nat,Y: set_nat] :
      ( ! [X: set_nat,Y2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y2)),X)
     => ( ! [X: set_nat,Y2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y2)),Y2)
       => ( ! [X: set_nat,Y2: set_nat,Z3: set_nat] :
              ( aa_set_nat_o(ord_less_eq_set_nat(X),Y2)
             => ( aa_set_nat_o(ord_less_eq_set_nat(X),Z3)
               => aa_set_nat_o(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z3)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Xa),Y) ) ) ) ) ).

% inf_unique
tff(fact_5567_inf__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,Xa: rat,Y: rat] :
      ( ! [X: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y2)),X)
     => ( ! [X: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y2)),Y2)
       => ( ! [X: rat,Y2: rat,Z3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(F,Y2),Z3)) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,Xa),Y) ) ) ) ) ).

% inf_unique
tff(fact_5568_inf__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,Xa: nat,Y: nat] :
      ( ! [X: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y2)),X)
     => ( ! [X: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y2)),Y2)
       => ( ! [X: nat,Y2: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z3)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,Xa),Y) ) ) ) ) ).

% inf_unique
tff(fact_5569_inf__unique,axiom,
    ! [F: fun_int_fun_int_int,Xa: int,Y: int] :
      ( ! [X: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X),Y2)),X)
     => ( ! [X: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X),Y2)),Y2)
       => ( ! [X: int,Y2: int,Z3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(F,Y2),Z3)) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y) = aa_int_int(aa_int_fun_int_int(F,Xa),Y) ) ) ) ) ).

% inf_unique
tff(fact_5570_inf_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
     => ord_less_eq_assn(A,B) ) ).

% inf.orderI
tff(fact_5571_inf_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) )
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B) ) ).

% inf.orderI
tff(fact_5572_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
     => aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% inf.orderI
tff(fact_5573_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% inf.orderI
tff(fact_5574_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% inf.orderI
tff(fact_5575_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% inf.orderI
tff(fact_5576_inf_OorderE,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
     => ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.orderE
tff(fact_5577_inf_OorderE,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
     => ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) ) ) ).

% inf.orderE
tff(fact_5578_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.orderE
tff(fact_5579_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.orderE
tff(fact_5580_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.orderE
tff(fact_5581_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.orderE
tff(fact_5582_le__infI2,axiom,
    ! [B: assn,Xa: assn,A: assn] :
      ( ord_less_eq_assn(B,Xa)
     => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),Xa) ) ).

% le_infI2
tff(fact_5583_le__infI2,axiom,
    ! [B: extended_enat,Xa: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),Xa)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),Xa) ) ).

% le_infI2
tff(fact_5584_le__infI2,axiom,
    ! [B: set_nat,Xa: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),Xa)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),Xa) ) ).

% le_infI2
tff(fact_5585_le__infI2,axiom,
    ! [B: rat,Xa: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),Xa) ) ).

% le_infI2
tff(fact_5586_le__infI2,axiom,
    ! [B: nat,Xa: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),Xa) ) ).

% le_infI2
tff(fact_5587_le__infI2,axiom,
    ! [B: int,Xa: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),Xa) ) ).

% le_infI2
tff(fact_5588_le__infI1,axiom,
    ! [A: assn,Xa: assn,B: assn] :
      ( ord_less_eq_assn(A,Xa)
     => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),Xa) ) ).

% le_infI1
tff(fact_5589_le__infI1,axiom,
    ! [A: extended_enat,Xa: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),Xa)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),Xa) ) ).

% le_infI1
tff(fact_5590_le__infI1,axiom,
    ! [A: set_nat,Xa: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),Xa)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),Xa) ) ).

% le_infI1
tff(fact_5591_le__infI1,axiom,
    ! [A: rat,Xa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),Xa) ) ).

% le_infI1
tff(fact_5592_le__infI1,axiom,
    ! [A: nat,Xa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),Xa) ) ).

% le_infI1
tff(fact_5593_le__infI1,axiom,
    ! [A: int,Xa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),Xa) ) ).

% le_infI1
tff(fact_5594_inf__mono,axiom,
    ! [A: assn,C: assn,B: assn,D: assn] :
      ( ord_less_eq_assn(A,C)
     => ( ord_less_eq_assn(B,D)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,C),D)) ) ) ).

% inf_mono
tff(fact_5595_inf__mono,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat,D: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),C)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),D)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,C),D)) ) ) ).

% inf_mono
tff(fact_5596_inf__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),C)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),D)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),D)) ) ) ).

% inf_mono
tff(fact_5597_inf__mono,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,C),D)) ) ) ).

% inf_mono
tff(fact_5598_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C),D)) ) ) ).

% inf_mono
tff(fact_5599_inf__mono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C),D)) ) ) ).

% inf_mono
tff(fact_5600_le__infI,axiom,
    ! [Xa: assn,A: assn,B: assn] :
      ( ord_less_eq_assn(Xa,A)
     => ( ord_less_eq_assn(Xa,B)
       => ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% le_infI
tff(fact_5601_le__infI,axiom,
    ! [Xa: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),A)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),B)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)) ) ) ).

% le_infI
tff(fact_5602_le__infI,axiom,
    ! [Xa: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),B)
       => aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% le_infI
tff(fact_5603_le__infI,axiom,
    ! [Xa: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)) ) ) ).

% le_infI
tff(fact_5604_le__infI,axiom,
    ! [Xa: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% le_infI
tff(fact_5605_le__infI,axiom,
    ! [Xa: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) ) ) ).

% le_infI
tff(fact_5606_le__infE,axiom,
    ! [Xa: assn,A: assn,B: assn] :
      ( ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B))
     => ~ ( ord_less_eq_assn(Xa,A)
         => ~ ord_less_eq_assn(Xa,B) ) ) ).

% le_infE
tff(fact_5607_le__infE,axiom,
    ! [Xa: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B))
     => ~ ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),A)
         => ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xa),B) ) ) ).

% le_infE
tff(fact_5608_le__infE,axiom,
    ! [Xa: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(Xa),A)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(Xa),B) ) ) ).

% le_infE
tff(fact_5609_le__infE,axiom,
    ! [Xa: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),B) ) ) ).

% le_infE
tff(fact_5610_le__infE,axiom,
    ! [Xa: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),B) ) ) ).

% le_infE
tff(fact_5611_le__infE,axiom,
    ! [Xa: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),B) ) ) ).

% le_infE
tff(fact_5612_inf__le2,axiom,
    ! [Xa: assn,Y: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y),Y) ).

% inf_le2
tff(fact_5613_inf__le2,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),Y) ).

% inf_le2
tff(fact_5614_inf__le2,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),Y) ).

% inf_le2
tff(fact_5615_inf__le2,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Y)),Y) ).

% inf_le2
tff(fact_5616_inf__le2,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),Y) ).

% inf_le2
tff(fact_5617_inf__le2,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),Y) ).

% inf_le2
tff(fact_5618_inf__le1,axiom,
    ! [Xa: assn,Y: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y),Xa) ).

% inf_le1
tff(fact_5619_inf__le1,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),Xa) ).

% inf_le1
tff(fact_5620_inf__le1,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),Xa) ).

% inf_le1
tff(fact_5621_inf__le1,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Y)),Xa) ).

% inf_le1
tff(fact_5622_inf__le1,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),Xa) ).

% inf_le1
tff(fact_5623_inf__le1,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),Xa) ).

% inf_le1
tff(fact_5624_inf__sup__ord_I1_J,axiom,
    ! [Xa: assn,Y: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y),Xa) ).

% inf_sup_ord(1)
tff(fact_5625_inf__sup__ord_I1_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),Xa) ).

% inf_sup_ord(1)
tff(fact_5626_inf__sup__ord_I1_J,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),Xa) ).

% inf_sup_ord(1)
tff(fact_5627_inf__sup__ord_I1_J,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Y)),Xa) ).

% inf_sup_ord(1)
tff(fact_5628_inf__sup__ord_I1_J,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),Xa) ).

% inf_sup_ord(1)
tff(fact_5629_inf__sup__ord_I1_J,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),Xa) ).

% inf_sup_ord(1)
tff(fact_5630_inf__sup__ord_I2_J,axiom,
    ! [Xa: assn,Y: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y),Y) ).

% inf_sup_ord(2)
tff(fact_5631_inf__sup__ord_I2_J,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_5632_inf__sup__ord_I2_J,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_5633_inf__sup__ord_I2_J,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_5634_inf__sup__ord_I2_J,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_5635_inf__sup__ord_I2_J,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_5636_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),C)
     => aa_set_nat_o(ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5637_inf_Ostrict__coboundedI2,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ord_less_assn(B,C)
     => ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),C) ) ).

% inf.strict_coboundedI2
tff(fact_5638_inf_Ostrict__coboundedI2,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),C)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5639_inf_Ostrict__coboundedI2,axiom,
    ! [B: real,C: real,A: real] :
      ( 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_real_real(inf_inf_real(A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5640_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5641_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5642_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5643_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),C)
     => aa_set_nat_o(ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5644_inf_Ostrict__coboundedI1,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ord_less_assn(A,C)
     => ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),C) ) ).

% inf.strict_coboundedI1
tff(fact_5645_inf_Ostrict__coboundedI1,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),C)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5646_inf_Ostrict__coboundedI1,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inf_inf_real(A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5647_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5648_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5649_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5650_inf_Ostrict__order__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5651_inf_Ostrict__order__iff,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_assn(A,B)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5652_inf_Ostrict__order__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),B)
    <=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5653_inf_Ostrict__order__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( ( A = aa_real_real(inf_inf_real(A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5654_inf_Ostrict__order__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5655_inf_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5656_inf_Ostrict__order__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5657_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
     => ~ ( aa_set_nat_o(ord_less_set_nat(A),B)
         => ~ aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5658_inf_Ostrict__boundedE,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ord_less_assn(A,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C))
     => ~ ( ord_less_assn(A,B)
         => ~ ord_less_assn(A,C) ) ) ).

% inf.strict_boundedE
tff(fact_5659_inf_Ostrict__boundedE,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,B),C))
     => ~ ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),B)
         => ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5660_inf_Ostrict__boundedE,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(inf_inf_real(B),C))
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5661_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5662_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5663_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5664_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_5665_inf_Oabsorb4,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_assn(B,A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb4
tff(fact_5666_inf_Oabsorb4,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_5667_inf_Oabsorb4,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(inf_inf_real(A),B) = B ) ) ).

% inf.absorb4
tff(fact_5668_inf_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_5669_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_5670_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb4
tff(fact_5671_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_5672_inf_Oabsorb3,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_assn(A,B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb3
tff(fact_5673_inf_Oabsorb3,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_5674_inf_Oabsorb3,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(inf_inf_real(A),B) = A ) ) ).

% inf.absorb3
tff(fact_5675_inf_Oabsorb3,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_5676_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_5677_inf_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb3
tff(fact_5678_less__infI2,axiom,
    ! [B: set_nat,Xa: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),Xa)
     => aa_set_nat_o(ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),Xa) ) ).

% less_infI2
tff(fact_5679_less__infI2,axiom,
    ! [B: assn,Xa: assn,A: assn] :
      ( ord_less_assn(B,Xa)
     => ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),Xa) ) ).

% less_infI2
tff(fact_5680_less__infI2,axiom,
    ! [B: extended_enat,Xa: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),Xa)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),Xa) ) ).

% less_infI2
tff(fact_5681_less__infI2,axiom,
    ! [B: real,Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inf_inf_real(A),B)),Xa) ) ).

% less_infI2
tff(fact_5682_less__infI2,axiom,
    ! [B: rat,Xa: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),Xa) ) ).

% less_infI2
tff(fact_5683_less__infI2,axiom,
    ! [B: nat,Xa: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),Xa) ) ).

% less_infI2
tff(fact_5684_less__infI2,axiom,
    ! [B: int,Xa: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),Xa) ) ).

% less_infI2
tff(fact_5685_less__infI1,axiom,
    ! [A: set_nat,Xa: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),Xa)
     => aa_set_nat_o(ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),Xa) ) ).

% less_infI1
tff(fact_5686_less__infI1,axiom,
    ! [A: assn,Xa: assn,B: assn] :
      ( ord_less_assn(A,Xa)
     => ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B),Xa) ) ).

% less_infI1
tff(fact_5687_less__infI1,axiom,
    ! [A: extended_enat,Xa: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),Xa)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),Xa) ) ).

% less_infI1
tff(fact_5688_less__infI1,axiom,
    ! [A: real,Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inf_inf_real(A),B)),Xa) ) ).

% less_infI1
tff(fact_5689_less__infI1,axiom,
    ! [A: rat,Xa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),Xa) ) ).

% less_infI1
tff(fact_5690_less__infI1,axiom,
    ! [A: nat,Xa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),Xa) ) ).

% less_infI1
tff(fact_5691_less__infI1,axiom,
    ! [A: int,Xa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),Xa) ) ).

% less_infI1
tff(fact_5692_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),Xa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),Xa)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_5693_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: assn,Z: assn,Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),Xa) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Xa)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),Xa)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_5694_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),Xa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),Xa)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_5695_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: assn,Z: assn,Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),Xa) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Xa)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),Xa)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_5696_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_5697_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_5698_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_5699_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_5700_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),Xa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_5701_sup__inf__distrib2,axiom,
    ! [Y: nat,Z: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_5702_sup__inf__distrib2,axiom,
    ! [Y: int,Z: int,Xa: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)),Xa) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Xa)),aa_int_int(aa_int_fun_int_int(sup_sup_int,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_5703_sup__inf__distrib2,axiom,
    ! [Y: extended_enat,Z: extended_enat,Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z)),Xa) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Xa)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_5704_sup__inf__distrib2,axiom,
    ! [Y: assn,Z: assn,Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),Xa) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Xa)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_5705_sup__inf__distrib1,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_5706_sup__inf__distrib1,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_5707_sup__inf__distrib1,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_5708_sup__inf__distrib1,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_5709_sup__inf__distrib1,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_5710_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),Xa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_5711_inf__sup__distrib2,axiom,
    ! [Y: nat,Z: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_5712_inf__sup__distrib2,axiom,
    ! [Y: int,Z: int,Xa: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),Xa) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Xa)),aa_int_int(aa_int_fun_int_int(inf_inf_int,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_5713_inf__sup__distrib2,axiom,
    ! [Y: extended_enat,Z: extended_enat,Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Z)),Xa) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Xa)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_5714_inf__sup__distrib2,axiom,
    ! [Y: assn,Z: assn,Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),Xa) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Xa)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_5715_inf__sup__distrib1,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_5716_inf__sup__distrib1,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_5717_inf__sup__distrib1,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_5718_inf__sup__distrib1,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Z)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_5719_inf__sup__distrib1,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_5720_distrib__imp2,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X: set_nat,Y2: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y2),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_5721_distrib__imp2,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ! [X: nat,Y2: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y2),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_5722_distrib__imp2,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( ! [X: int,Y2: int,Z3: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y2),Z3)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z3))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_5723_distrib__imp2,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ! [X: extended_enat,Y2: extended_enat,Z3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,X),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y2),Z3)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,X),Y2)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,X),Z3))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Z)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_5724_distrib__imp2,axiom,
    ! [Xa: assn,Y: assn,Z: assn] :
      ( ! [X: assn,Y2: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y2),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z3))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_5725_distrib__imp1,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X: set_nat,Y2: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y2),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_5726_distrib__imp1,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ! [X: nat,Y2: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y2),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_5727_distrib__imp1,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( ! [X: int,Y2: int,Z3: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y2),Z3)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y2)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z3))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_5728_distrib__imp1,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ! [X: extended_enat,Y2: extended_enat,Z3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,X),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y2),Z3)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,X),Y2)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,X),Z3))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_5729_distrib__imp1,axiom,
    ! [Xa: assn,Y: assn,Z: assn] :
      ( ! [X: assn,Y2: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y2),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z3))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_5730_disjointI,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A)
         => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B) )
     => ( inf_in473752961466201218T_VEBT(A,B) = bot_bo8194388402131092736T_VEBT ) ) ).

% disjointI
tff(fact_5731_disjointI,axiom,
    ! [A: set_real,B: set_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A)
         => ~ aa_set_real_o(member_real(X),B) )
     => ( inf_inf_set_real(A,B) = bot_bot_set_real ) ) ).

% disjointI
tff(fact_5732_disjointI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A)
         => ~ aa_set_nat_o(member_nat(X),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ).

% disjointI
tff(fact_5733_disjointI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A)
         => ~ aa_set_int_o(member_int(X),B) )
     => ( inf_inf_set_int(A,B) = bot_bot_set_int ) ) ).

% disjointI
tff(fact_5734_disjoint__iff__not__equal,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),A2)
         => ! [Xa3: real] :
              ( aa_set_real_o(member_real(Xa3),B2)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_5735_disjoint__iff__not__equal,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A2)
         => ! [Xa3: nat] :
              ( aa_set_nat_o(member_nat(Xa3),B2)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_5736_disjoint__iff__not__equal,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(A2,B2) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A2)
         => ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),B2)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_5737_Int__empty__right,axiom,
    ! [A2: set_real] : inf_inf_set_real(A2,bot_bot_set_real) = bot_bot_set_real ).

% Int_empty_right
tff(fact_5738_Int__empty__right,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_5739_Int__empty__right,axiom,
    ! [A2: set_int] : inf_inf_set_int(A2,bot_bot_set_int) = bot_bot_set_int ).

% Int_empty_right
tff(fact_5740_Int__empty__left,axiom,
    ! [B2: set_real] : inf_inf_set_real(bot_bot_set_real,B2) = bot_bot_set_real ).

% Int_empty_left
tff(fact_5741_Int__empty__left,axiom,
    ! [B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B2) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_5742_Int__empty__left,axiom,
    ! [B2: set_int] : inf_inf_set_int(bot_bot_set_int,B2) = bot_bot_set_int ).

% Int_empty_left
tff(fact_5743_disjoint__iff,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( inf_in473752961466201218T_VEBT(A2,B2) = bot_bo8194388402131092736T_VEBT )
    <=> ! [X4: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),A2)
         => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),B2) ) ) ).

% disjoint_iff
tff(fact_5744_disjoint__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),A2)
         => ~ aa_set_real_o(member_real(X4),B2) ) ) ).

% disjoint_iff
tff(fact_5745_disjoint__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A2)
         => ~ aa_set_nat_o(member_nat(X4),B2) ) ) ).

% disjoint_iff
tff(fact_5746_disjoint__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(A2,B2) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A2)
         => ~ aa_set_int_o(member_int(X4),B2) ) ) ).

% disjoint_iff
tff(fact_5747_Int__emptyI,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
         => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B2) )
     => ( inf_in473752961466201218T_VEBT(A2,B2) = bot_bo8194388402131092736T_VEBT ) ) ).

% Int_emptyI
tff(fact_5748_Int__emptyI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => ~ aa_set_real_o(member_real(X),B2) )
     => ( inf_inf_set_real(A2,B2) = bot_bot_set_real ) ) ).

% Int_emptyI
tff(fact_5749_Int__emptyI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => ~ aa_set_nat_o(member_nat(X),B2) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_5750_Int__emptyI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => ~ aa_set_int_o(member_int(X),B2) )
     => ( inf_inf_set_int(A2,B2) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_5751_Int__mono,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat,D3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),D3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D3)) ) ) ).

% Int_mono
tff(fact_5752_Int__lower1,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),A2) ).

% Int_lower1
tff(fact_5753_Int__lower2,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),B2) ).

% Int_lower2
tff(fact_5754_Int__absorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_5755_Int__absorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = A2 ) ) ).

% Int_absorb2
tff(fact_5756_Int__greatest,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C2),A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C2),B2)
       => aa_set_nat_o(ord_less_eq_set_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% Int_greatest
tff(fact_5757_Int__Collect__mono,axiom,
    ! [A2: set_real,B2: set_real,P: fun_real_o,Q: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),A2)
           => ( aa_real_o(P,X)
             => aa_real_o(Q,X) ) )
       => aa_set_real_o(ord_less_eq_set_real(inf_inf_set_real(A2,collect_real(P))),inf_inf_set_real(B2,collect_real(Q))) ) ) ).

% Int_Collect_mono
tff(fact_5758_Int__Collect__mono,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,P: fun_VEBT_VEBT_o,Q: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B2)
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
           => ( aa_VEBT_VEBT_o(P,X)
             => aa_VEBT_VEBT_o(Q,X) ) )
       => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(inf_in473752961466201218T_VEBT(A2,collect_VEBT_VEBT(P))),inf_in473752961466201218T_VEBT(B2,collect_VEBT_VEBT(Q))) ) ) ).

% Int_Collect_mono
tff(fact_5759_Int__Collect__mono,axiom,
    ! [A2: set_int,B2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A2)
           => ( aa_int_o(P,X)
             => aa_int_o(Q,X) ) )
       => aa_set_int_o(ord_less_eq_set_int(inf_inf_set_int(A2,collect_int(P))),inf_inf_set_int(B2,collect_int(Q))) ) ) ).

% Int_Collect_mono
tff(fact_5760_Int__Collect__mono,axiom,
    ! [A2: set_complex,B2: set_complex,P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A2)
           => ( aa_complex_o(P,X)
             => aa_complex_o(Q,X) ) )
       => aa_set_complex_o(ord_le211207098394363844omplex(inf_inf_set_complex(A2,collect_complex(P))),inf_inf_set_complex(B2,collect_complex(Q))) ) ) ).

% Int_Collect_mono
tff(fact_5761_Int__Collect__mono,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int,P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(A2,B2)
     => ( ! [X: product_prod_int_int] :
            ( member5262025264175285858nt_int(X,A2)
           => ( aa_Pro4592068669816069595_int_o(P,X)
             => aa_Pro4592068669816069595_int_o(Q,X) ) )
       => ord_le2843351958646193337nt_int(inf_in2269163501485487111nt_int(A2,collec213857154873943460nt_int(P)),inf_in2269163501485487111nt_int(B2,collec213857154873943460nt_int(Q))) ) ) ).

% Int_Collect_mono
tff(fact_5762_Int__Collect__mono,axiom,
    ! [A2: set_nat,B2: set_nat,P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A2)
           => ( aa_nat_o(P,X)
             => aa_nat_o(Q,X) ) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),collect_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_5763_Int__insert__right,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      inf_inf_set_real(A2,insert_real(A,B2)) = $ite(aa_set_real_o(member_real(A),A2),insert_real(A,inf_inf_set_real(A2,B2)),inf_inf_set_real(A2,B2)) ).

% Int_insert_right
tff(fact_5764_Int__insert__right,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      inf_inf_set_int(A2,insert_int(A,B2)) = $ite(aa_set_int_o(member_int(A),A2),insert_int(A,inf_inf_set_int(A2,B2)),inf_inf_set_int(A2,B2)) ).

% Int_insert_right
tff(fact_5765_Int__insert__right,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
      inf_in473752961466201218T_VEBT(A2,insert_VEBT_VEBT(A,B2)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2),insert_VEBT_VEBT(A,inf_in473752961466201218T_VEBT(A2,B2)),inf_in473752961466201218T_VEBT(A2,B2)) ).

% Int_insert_right
tff(fact_5766_Int__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat(A,B2)) = $ite(aa_set_nat_o(member_nat(A),A2),insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ).

% Int_insert_right
tff(fact_5767_Int__insert__left,axiom,
    ! [A: real,B2: set_real,C2: set_real] :
      inf_inf_set_real(insert_real(A,B2),C2) = $ite(aa_set_real_o(member_real(A),C2),insert_real(A,inf_inf_set_real(B2,C2)),inf_inf_set_real(B2,C2)) ).

% Int_insert_left
tff(fact_5768_Int__insert__left,axiom,
    ! [A: int,B2: set_int,C2: set_int] :
      inf_inf_set_int(insert_int(A,B2),C2) = $ite(aa_set_int_o(member_int(A),C2),insert_int(A,inf_inf_set_int(B2,C2)),inf_inf_set_int(B2,C2)) ).

% Int_insert_left
tff(fact_5769_Int__insert__left,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,C2: set_VEBT_VEBT] :
      inf_in473752961466201218T_VEBT(insert_VEBT_VEBT(A,B2),C2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),C2),insert_VEBT_VEBT(A,inf_in473752961466201218T_VEBT(B2,C2)),inf_in473752961466201218T_VEBT(B2,C2)) ).

% Int_insert_left
tff(fact_5770_Int__insert__left,axiom,
    ! [A: nat,B2: set_nat,C2: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B2)),C2) = $ite(aa_set_nat_o(member_nat(A),C2),insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ).

% Int_insert_left
tff(fact_5771_Diff__Int__distrib2,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ).

% Diff_Int_distrib2
tff(fact_5772_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),B2)) ).

% Diff_Int_distrib
tff(fact_5773_Diff__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% Diff_Diff_Int
tff(fact_5774_Diff__Int2,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)),B2) ).

% Diff_Int2
tff(fact_5775_Int__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C2)) ).

% Int_Diff
tff(fact_5776_Un__Int__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A2)) ).

% Un_Int_distrib2
tff(fact_5777_Int__Un__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A2)) ).

% Int_Un_distrib2
tff(fact_5778_Un__Int__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C2)) ).

% Un_Int_distrib
tff(fact_5779_Int__Un__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)) ).

% Int_Un_distrib
tff(fact_5780_Un__Int__crazy,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A2)) ).

% Un_Int_crazy
tff(fact_5781_distrib__inf__le,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Y)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),Z))),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Y),Z))) ).

% distrib_inf_le
tff(fact_5782_distrib__inf__le,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Z)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z))) ).

% distrib_inf_le
tff(fact_5783_distrib__inf__le,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_5784_distrib__inf__le,axiom,
    ! [Xa: rat,Y: rat,Z: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z))) ).

% distrib_inf_le
tff(fact_5785_distrib__inf__le,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_5786_distrib__inf__le,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z))) ).

% distrib_inf_le
tff(fact_5787_distrib__sup__le,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Y),Z))),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Z))) ).

% distrib_sup_le
tff(fact_5788_distrib__sup__le,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Z))) ).

% distrib_sup_le
tff(fact_5789_distrib__sup__le,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z))) ).

% distrib_sup_le
tff(fact_5790_distrib__sup__le,axiom,
    ! [Xa: rat,Y: rat,Z: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Y)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Z))) ).

% distrib_sup_le
tff(fact_5791_distrib__sup__le,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z))) ).

% distrib_sup_le
tff(fact_5792_distrib__sup__le,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Z))) ).

% distrib_sup_le
tff(fact_5793_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na),set_or1266510415728281911st_int(M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_5794_disjoint__mono,axiom,
    ! [A: set_real,A3: set_real,B: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),A3)
     => ( aa_set_real_o(ord_less_eq_set_real(B),B4)
       => ( ( inf_inf_set_real(A3,B4) = bot_bot_set_real )
         => ( inf_inf_set_real(A,B) = bot_bot_set_real ) ) ) ) ).

% disjoint_mono
tff(fact_5795_disjoint__mono,axiom,
    ! [A: set_int,A3: set_int,B: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),A3)
     => ( aa_set_int_o(ord_less_eq_set_int(B),B4)
       => ( ( inf_inf_set_int(A3,B4) = bot_bot_set_int )
         => ( inf_inf_set_int(A,B) = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
tff(fact_5796_disjoint__mono,axiom,
    ! [A: set_nat,A3: set_nat,B: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),A3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B4) = bot_bot_set_nat )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
tff(fact_5797_disjoint__alt__simp2,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A2,B2) != A2 )
    <=> ( inf_inf_set_real(A2,B2) != bot_bot_set_real ) ) ).

% disjoint_alt_simp2
tff(fact_5798_disjoint__alt__simp2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) != A2 )
    <=> ( inf_inf_set_int(A2,B2) != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
tff(fact_5799_disjoint__alt__simp2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) != A2 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
tff(fact_5800_disjoint__alt__simp1,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A2,B2) = A2 )
    <=> ( inf_inf_set_real(A2,B2) = bot_bot_set_real ) ) ).

% disjoint_alt_simp1
tff(fact_5801_disjoint__alt__simp1,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) = A2 )
    <=> ( inf_inf_set_int(A2,B2) = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
tff(fact_5802_disjoint__alt__simp1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = A2 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
tff(fact_5803_Diff__triv,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
     => ( minus_minus_set_real(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_5804_Diff__triv,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(A2,B2) = bot_bot_set_int )
     => ( minus_minus_set_int(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_5805_Diff__triv,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = A2 ) ) ).

% Diff_triv
tff(fact_5806_Int__Diff__disjoint,axiom,
    ! [A2: set_real,B2: set_real] : inf_inf_set_real(inf_inf_set_real(A2,B2),minus_minus_set_real(A2,B2)) = bot_bot_set_real ).

% Int_Diff_disjoint
tff(fact_5807_Int__Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] : inf_inf_set_int(inf_inf_set_int(A2,B2),minus_minus_set_int(A2,B2)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_5808_Int__Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_5809_Un__Int__assoc__eq,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) )
    <=> aa_set_nat_o(ord_less_eq_set_nat(C2),A2) ) ).

% Un_Int_assoc_eq
tff(fact_5810_Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C2)) ).

% Diff_Un
tff(fact_5811_Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C2)) ).

% Diff_Int
tff(fact_5812_Int__Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = A2 ).

% Int_Diff_Un
tff(fact_5813_Un__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = A2 ).

% Un_Diff_Int
tff(fact_5814_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_5815_bset_I1_J,axiom,
    ! [D3: int,B2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),B2)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
         => ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B2)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
           => ( aa_int_o(Q,X)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( aa_int_o(P,X5)
                & aa_int_o(Q,X5) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)) ) ) ) ) ) ).

% bset(1)
tff(fact_5816_bset_I2_J,axiom,
    ! [D3: int,B2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),B2)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
         => ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B2)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
           => ( aa_int_o(Q,X)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( aa_int_o(P,X5)
                | aa_int_o(Q,X5) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)) ) ) ) ) ) ).

% bset(2)
tff(fact_5817_aset_I1_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),A2)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
         => ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A2)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
           => ( aa_int_o(Q,X)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( aa_int_o(P,X5)
                & aa_int_o(Q,X5) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)) ) ) ) ) ) ).

% aset(1)
tff(fact_5818_aset_I2_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),A2)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
         => ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A2)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
           => ( aa_int_o(Q,X)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( aa_int_o(P,X5)
                | aa_int_o(Q,X5) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)) ) ) ) ) ) ).

% aset(2)
tff(fact_5819_disjoint__alt__simp3,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(minus_minus_set_real(A2,B2)),A2)
    <=> ( inf_inf_set_real(A2,B2) != bot_bot_set_real ) ) ).

% disjoint_alt_simp3
tff(fact_5820_disjoint__alt__simp3,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(minus_minus_set_int(A2,B2)),A2)
    <=> ( inf_inf_set_int(A2,B2) != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
tff(fact_5821_disjoint__alt__simp3,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),A2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
tff(fact_5822_bset_I3_J,axiom,
    ! [D3: int,T: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( X5 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3) = T ) ) ) ) ) ).

% bset(3)
tff(fact_5823_bset_I4_J,axiom,
    ! [D3: int,T: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T),B2)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( X5 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3) != T ) ) ) ) ) ).

% bset(4)
tff(fact_5824_bset_I5_J,axiom,
    ! [D3: int,B2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)),T) ) ) ) ).

% bset(5)
tff(fact_5825_bset_I7_J,axiom,
    ! [D3: int,T: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T),B2)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)) ) ) ) ) ).

% bset(7)
tff(fact_5826_aset_I3_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( X5 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3) = T ) ) ) ) ) ).

% aset(3)
tff(fact_5827_aset_I4_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T),A2)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( X5 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3) != T ) ) ) ) ) ).

% aset(4)
tff(fact_5828_aset_I5_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T),A2)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)),T) ) ) ) ) ).

% aset(5)
tff(fact_5829_aset_I7_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)) ) ) ) ).

% aset(7)
tff(fact_5830_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)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P,X)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [X_12: int] : aa_int_o(P,X_12)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5831_bset_I6_J,axiom,
    ! [D3: int,B2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)),T) ) ) ) ).

% bset(6)
tff(fact_5832_bset_I8_J,axiom,
    ! [D3: int,T: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D3)) ) ) ) ) ).

% bset(8)
tff(fact_5833_aset_I6_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)),T) ) ) ) ) ).

% aset(6)
tff(fact_5834_aset_I8_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D3)) ) ) ) ).

% aset(8)
tff(fact_5835_cpmi,axiom,
    ! [D3: int,P: fun_int_o,P2: fun_int_o,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z5: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z5)
           => ( aa_int_o(P,X)
            <=> aa_int_o(P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb: int] :
                      ( aa_set_int_o(member_int(Xb),B2)
                     => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
             => ( aa_int_o(P,X)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)) ) )
         => ( ! [X: int,K2: int] :
                ( aa_int_o(P2,X)
              <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P2,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [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),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5836_cppi,axiom,
    ! [D3: int,P: fun_int_o,P2: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z5: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X)
           => ( aa_int_o(P,X)
            <=> aa_int_o(P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb: int] :
                      ( aa_set_int_o(member_int(Xb),A2)
                     => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
             => ( aa_int_o(P,X)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)) ) )
         => ( ! [X: int,K2: int] :
                ( aa_int_o(P2,X)
              <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P2,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5837_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),U)
       => ( sup_sup_set_rat(set_or633870826150836451st_rat(L,M),set_or633870826150836451st_rat(M,U)) = set_or633870826150836451st_rat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5838_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),U)
       => ( sup_sup_set_num(set_or7049704709247886629st_num(L,M),set_or7049704709247886629st_num(M,U)) = set_or7049704709247886629st_num(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5839_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5840_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),U)
       => ( sup_sup_set_int(set_or1266510415728281911st_int(L,M),set_or1266510415728281911st_int(M,U)) = set_or1266510415728281911st_int(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5841_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: real,M: real,U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M),U)
       => ( sup_sup_set_real(set_or1222579329274155063t_real(L,M),set_or1222579329274155063t_real(M,U)) = set_or1222579329274155063t_real(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5842_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = insert_nat(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5843_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( set_or1266510415728281911st_int(A,B) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5844_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( set_or1222579329274155063t_real(A,B) = insert_real(A,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5845_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(set_or189985376899183464nteger(A,B)),set_or189985376899183464nteger(C,D))
    <=> ( ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
          | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D)
            & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A)
              | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),D) ) ) )
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5846_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
          | ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
            & aa_set_nat_o(ord_less_eq_set_nat(B),D)
            & ( aa_set_nat_o(ord_less_set_nat(C),A)
              | aa_set_nat_o(ord_less_set_nat(B),D) ) ) )
        & aa_set_nat_o(ord_less_eq_set_nat(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5847_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5848_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5849_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5850_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5851_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5852_time__replicate,axiom,
    ! [Xa: heap_T8145700208782473153_VEBTi,C: nat,Na: nat,H: heap_e7401611519738050253t_unit] :
      ( ! [H2: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(Xa,H2)),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t3534373299052942712_VEBTi(vEBT_V1859673955506687831_VEBTi(Na,Xa),H)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Na))) ) ).

% time_replicate
tff(fact_5853_TBOUND__buildupi,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_vebt_buildupi(Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% TBOUND_buildupi
tff(fact_5854_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(T,Xa))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),Xa)),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_5855_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(T),Xa)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(T,Y)),Xa) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_5856_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(T,Xa)),Xa) ) ) ).

% valid_insert_both_member_options_add
tff(fact_5857_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(T,Xa))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(T),Xa)),one_one_nat) ) ) ).

% minNull_delete_time_bound'
tff(fact_5858_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(T),X_1) ) ).

% not_min_Null_member
tff(fact_5859_min__Null__member,axiom,
    ! [T: vEBT_VEBT,Xa: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ~ aa_nat_o(vEBT_vebt_member(T),Xa) ) ).

% min_Null_member
tff(fact_5860_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_delete(T,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa_nat_o(vEBT_V8194947554948674370ptions(T),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_5861_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(T),Xa)
       => aa_nat_o(vEBT_vebt_member(T),Xa) ) ) ).

% valid_member_both_member_options
tff(fact_5862_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(T),Xa)
      <=> aa_nat_o(vEBT_vebt_member(T),Xa) ) ) ).

% both_member_options_equiv_member
tff(fact_5863_TBOUND__vebt__buildupi,axiom,
    ! [Na: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V739175172307565963ildupi(Na)),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% TBOUND_vebt_buildupi
tff(fact_5864_set__vebt__minNull,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
      <=> ( vEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% set_vebt_minNull
tff(fact_5865_TBOUND__replicate,axiom,
    ! [Xa: heap_T8145700208782473153_VEBTi,C: nat,Na: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(Xa),C)
     => aa_nat_o(time_T8149879359713347829_VEBTi(vEBT_V1859673955506687831_VEBTi(Na,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Na))) ) ).

% TBOUND_replicate
tff(fact_5866_TBOUND__replicate,axiom,
    ! [Xa: heap_Time_Heap_o,C: nat,Na: nat] :
      ( aa_nat_o(time_TBOUND_o(Xa),C)
     => aa_nat_o(time_TBOUND_list_o(vEBT_V2326993469660664182atei_o(Na,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Na))) ) ).

% TBOUND_replicate
tff(fact_5867_TBOUND__replicate,axiom,
    ! [Xa: heap_T2636463487746394924on_nat,C: nat,Na: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(Xa),C)
     => aa_nat_o(time_T3808005469503390304on_nat(vEBT_V792416675989592002on_nat(Na,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Na))) ) ).

% TBOUND_replicate
tff(fact_5868_TBOUND__replicate,axiom,
    ! [Xa: heap_Time_Heap_nat,C: nat,Na: nat] :
      ( aa_nat_o(time_TBOUND_nat(Xa),C)
     => aa_nat_o(time_TBOUND_list_nat(vEBT_V7726092123322077554ei_nat(Na,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),Na))) ) ).

% TBOUND_replicate
tff(fact_5869_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(T,Tia,Xa)),one_one_nat) ) ).

% TBOUND_minNull
tff(fact_5870_TBOUND__vebt__deletei,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V1365221501068881998eletei(T,Tia,Xa)),
        $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(T,Xa)),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))))) ).

% TBOUND_vebt_deletei
tff(fact_5871_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(T,Tia,Xa)),
        $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))))) ).

% TBOUND_vebt_inserti
tff(fact_5872_inf__Int__eq,axiom,
    ! [R2: set_real,S: set_real,X5: real] :
      ( aa_real_o(inf_inf_fun_real_o(aTP_Lamm_a(R2),aTP_Lamm_a(S)),X5)
    <=> aa_set_real_o(member_real(X5),inf_inf_set_real(R2,S)) ) ).

% inf_Int_eq
tff(fact_5873_inf__Int__eq,axiom,
    ! [R2: set_int,S: set_int,X5: int] :
      ( aa_int_o(inf_inf_fun_int_o(aTP_Lamm_ac(R2),aTP_Lamm_ac(S)),X5)
    <=> aa_set_int_o(member_int(X5),inf_inf_set_int(R2,S)) ) ).

% inf_Int_eq
tff(fact_5874_inf__Int__eq,axiom,
    ! [R2: set_VEBT_VEBT,S: set_VEBT_VEBT,X5: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(inf_in3591633114662893881VEBT_o(aTP_Lamm_aa(R2),aTP_Lamm_aa(S)),X5)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),inf_in473752961466201218T_VEBT(R2,S)) ) ).

% inf_Int_eq
tff(fact_5875_inf__Int__eq,axiom,
    ! [R2: set_nat,S: set_nat,X5: nat] :
      ( aa_nat_o(inf_inf_fun_nat_o(aTP_Lamm_ab(R2),aTP_Lamm_ab(S)),X5)
    <=> aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,R2),S)) ) ).

% inf_Int_eq
tff(fact_5876_inf__set__def,axiom,
    ! [A2: set_real,B2: set_real] : inf_inf_set_real(A2,B2) = collect_real(inf_inf_fun_real_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).

% inf_set_def
tff(fact_5877_inf__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : inf_in473752961466201218T_VEBT(A2,B2) = collect_VEBT_VEBT(inf_in3591633114662893881VEBT_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% inf_set_def
tff(fact_5878_inf__set__def,axiom,
    ! [A2: set_int,B2: set_int] : inf_inf_set_int(A2,B2) = collect_int(inf_inf_fun_int_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).

% inf_set_def
tff(fact_5879_inf__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] : inf_inf_set_complex(A2,B2) = collect_complex(inf_in8039933884210822705plex_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2))) ).

% inf_set_def
tff(fact_5880_inf__set__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : inf_in2269163501485487111nt_int(A2,B2) = collec213857154873943460nt_int(inf_in1782598092406300404_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% inf_set_def
tff(fact_5881_inf__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = collect_nat(inf_inf_fun_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% inf_set_def
tff(fact_5882_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(vEBT_V8194947554948674370ptions(T)) ).

% set_vebt_def
tff(fact_5883_insersimp,axiom,
    ! [T: vEBT_VEBT,Na: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(T),X_1)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insersimp
tff(fact_5884_insertsimp,axiom,
    ! [T: vEBT_VEBT,Na: nat,L: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insertsimp
tff(fact_5885_buildup__nothing__in__leaf,axiom,
    ! [Na: nat,Xa: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Na)),Xa) ).

% buildup_nothing_in_leaf
tff(fact_5886_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => 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(T),Xa))),aa_real_real(aa_rea7340605660300375871l_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_5887_round__unique,axiom,
    ! [Xa: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xa),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,Y))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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,Xa) = Y ) ) ) ).

% round_unique
tff(fact_5888_round__unique,axiom,
    ! [Xa: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,Xa) = Y ) ) ) ).

% round_unique
tff(fact_5889_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => 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(T),Xa))),aa_real_real(aa_rea7340605660300375871l_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_5890_vebt__buildupi__rule,axiom,
    ! [Na: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_o_assn(pure_assn,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot_set_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% vebt_buildupi_rule
tff(fact_5891_tanh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_5892_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xa),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X4) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_5893_tanh__real__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(tanh_real,Xa) = aa_real_real(tanh_real,Y) )
    <=> ( Xa = Y ) ) ).

% tanh_real_eq_iff
tff(fact_5894_tanh__0,axiom,
    tanh_complex(zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_5895_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_5896_tanh__real__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(tanh_real,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% tanh_real_zero_iff
tff(fact_5897_tanh__real__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% tanh_real_less_iff
tff(fact_5898_tanh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xa)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% tanh_real_le_iff
tff(fact_5899_round__of__int,axiom,
    ! [Na: int] : aa_real_int(archim8280529875227126926d_real,aa_int_real(ring_1_of_int_real,Na)) = Na ).

% round_of_int
tff(fact_5900_round__of__int,axiom,
    ! [Na: int] : aa_rat_int(archim7778729529865785530nd_rat,aa_int_rat(ring_1_of_int_rat,Na)) = Na ).

% round_of_int
tff(fact_5901_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5902_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_5903_round__numeral,axiom,
    ! [Na: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% round_numeral
tff(fact_5904_round__numeral,axiom,
    ! [Na: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% round_numeral
tff(fact_5905_tanh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% tanh_real_pos_iff
tff(fact_5906_tanh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_5907_tanh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_5908_tanh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% tanh_real_nonneg_iff
tff(fact_5909_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_5910_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_5911_round__of__nat,axiom,
    ! [Na: nat] : aa_real_int(archim8280529875227126926d_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% round_of_nat
tff(fact_5912_round__of__nat,axiom,
    ! [Na: nat] : aa_rat_int(archim7778729529865785530nd_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% round_of_nat
tff(fact_5913_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_TBOUND_o(vEBT_V854960066525838166emberi(T,Tia,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% TBOUND_vebt_memberi
tff(fact_5914_TBOUND__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_VEBT_vebt_succi(T,Tia,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% TBOUND_vebt_succi
tff(fact_5915_TBOUND__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_VEBT_vebt_predi(T,Tia,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% TBOUND_vebt_predi
tff(fact_5916_artanh__tanh__real,axiom,
    ! [Xa: real] : aa_real_real(artanh_real,aa_real_real(tanh_real,Xa)) = Xa ).

% artanh_tanh_real
tff(fact_5917_tanh__real__lt__1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),one_one_real) ).

% tanh_real_lt_1
tff(fact_5918_round__mono,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xa)),aa_rat_int(archim7778729529865785530nd_rat,Y)) ) ).

% round_mono
tff(fact_5919_ceiling__ge__round,axiom,
    ! [Xa: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,Xa)),aa_real_int(archim7802044766580827645g_real,Xa)) ).

% ceiling_ge_round
tff(fact_5920_ceiling__ge__round,axiom,
    ! [Xa: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xa)),aa_rat_int(archim2889992004027027881ng_rat,Xa)) ).

% ceiling_ge_round
tff(fact_5921_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d2(T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% pred_bound_height'
tff(fact_5922_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c2(T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% succ'_bound_height
tff(fact_5923_of__int__round__le,axiom,
    ! [Xa: 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,Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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_5924_of__int__round__le,axiom,
    ! [Xa: 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,Xa))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_5925_of__int__round__ge,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Xa),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,Xa))) ).

% of_int_round_ge
tff(fact_5926_of__int__round__ge,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))) ).

% of_int_round_ge
tff(fact_5927_of__int__round__gt,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xa),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,Xa))) ).

% of_int_round_gt
tff(fact_5928_of__int__round__gt,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))) ).

% of_int_round_gt
tff(fact_5929_VEBT__internal_Ovebt__buildupi__rule,axiom,
    ! [Na: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_o_assn(pure_assn,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)),vEBT_vebt_buildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% VEBT_internal.vebt_buildupi_rule
tff(fact_5930_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_memberi(Tia,Xa),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_di(T),Tia),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,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_5931_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Na)
     => ( aa_nat_o(vEBT_vebt_member(Tree),Xa)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xa)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),Xa) ) ) ) ).

% member_valid_both_member_options
tff(fact_5932_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_memberi(Tia,Xa),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_di(T),Tia),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,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% htt_vebt_memberi
tff(fact_5933_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : aa_nat_o(time_TBOUND_o(vEBT_VEBT_minNulli(T)),one_one_nat) ).

% TBOUND_minNulli
tff(fact_5934_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(T),Xa)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(T),Xa)
        | aa_nat_o(vEBT_VEBT_membermima(T),Xa) ) ) ).

% both_member_options_def
tff(fact_5935_vebt__deletei__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine5565527176597971370_VEBTi(vEBT_vebt_deletei(Tia,Xa),vEBT_V1365221501068881998eletei(T,Tia,Xa)) ).

% vebt_deletei_refines
tff(fact_5936_buildup__nothing__in__min__max,axiom,
    ! [Na: nat,Xa: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Na)),Xa) ).

% buildup_nothing_in_min_max
tff(fact_5937_refines__replicate,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,F2: heap_T8145700208782473153_VEBTi,Na: nat] :
      ( refine5565527176597971370_VEBTi(F,F2)
     => refine3700189196150522554_VEBTi(vEBT_V1859673955506687831_VEBTi(Na,F),vEBT_V1859673955506687831_VEBTi(Na,F2)) ) ).

% refines_replicate
tff(fact_5938_refines__replicate,axiom,
    ! [F: heap_Time_Heap_o,F2: heap_Time_Heap_o,Na: nat] :
      ( refine_Imp_refines_o(F,F2)
     => refine5896690332125372649list_o(vEBT_V2326993469660664182atei_o(Na,F),vEBT_V2326993469660664182atei_o(Na,F2)) ) ).

% refines_replicate
tff(fact_5939_refines__replicate,axiom,
    ! [F: heap_T2636463487746394924on_nat,F2: heap_T2636463487746394924on_nat,Na: nat] :
      ( refine7594492741263601813on_nat(F,F2)
     => refine1935026298455697829on_nat(vEBT_V792416675989592002on_nat(Na,F),vEBT_V792416675989592002on_nat(Na,F2)) ) ).

% refines_replicate
tff(fact_5940_vebt__buildupi__refines,axiom,
    ! [Na: nat] : refine5565527176597971370_VEBTi(vEBT_vebt_buildupi(Na),vEBT_V739175172307565963ildupi(Na)) ).

% vebt_buildupi_refines
tff(fact_5941_vebt__inserti__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine5565527176597971370_VEBTi(vEBT_vebt_inserti(Tia,Xa),vEBT_V3964819847710782039nserti(T,Tia,Xa)) ).

% vebt_inserti_refines
tff(fact_5942_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_minNulli(Tia),aa_VEB6687300313434771872o_assn(aTP_Lamm_dj(T),Tia)),one_one_nat) ).

% minNrulli_ruleT
tff(fact_5943_refines__case__VEBTi,axiom,
    ! [Tia: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_6972675622308511895_VEBTi,F12: fun_o_6972675622308511895_VEBTi,F22: fun_op5127434927081926045_VEBTi,F23: fun_op5127434927081926045_VEBTi] :
      ( ( Tia = Ti )
     => ( ! [A4: $o,B5: $o] : refine5565527176597971370_VEBTi(aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F1,(A4)),(B5)),aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F12,(A4)),(B5)))
       => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F22,Info),Deg2),TreeArray),Summary),aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F23,Info),Deg2),TreeArray),Summary))
         => refine5565527176597971370_VEBTi(vEBT_c6028912655521741485_VEBTi(F22,F1,Tia),vEBT_c6028912655521741485_VEBTi(F23,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5944_refines__case__VEBTi,axiom,
    ! [Tia: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_1207901889202581752Heap_o,F12: fun_o_1207901889202581752Heap_o,F22: fun_op1763656958002597022Heap_o,F23: fun_op1763656958002597022Heap_o] :
      ( ( Tia = Ti )
     => ( ! [A4: $o,B5: $o] : refine_Imp_refines_o(aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F1,(A4)),(B5)),aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F12,(A4)),(B5)))
       => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine_Imp_refines_o(aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F22,Info),Deg2),TreeArray),Summary),aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F23,Info),Deg2),TreeArray),Summary))
         => refine_Imp_refines_o(vEBT_c6104975476656191286Heap_o(F22,F1,Tia),vEBT_c6104975476656191286Heap_o(F23,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5945_refines__case__VEBTi,axiom,
    ! [Tia: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_4049874799390722306on_nat,F12: fun_o_4049874799390722306on_nat,F22: fun_op5783775701855621896on_nat,F23: fun_op5783775701855621896on_nat] :
      ( ( Tia = Ti )
     => ( ! [A4: $o,B5: $o] : refine7594492741263601813on_nat(aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F1,(A4)),(B5)),aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F12,(A4)),(B5)))
       => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine7594492741263601813on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F22,Info),Deg2),TreeArray),Summary),aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F23,Info),Deg2),TreeArray),Summary))
         => refine7594492741263601813on_nat(vEBT_c6250501799366334488on_nat(F22,F1,Tia),vEBT_c6250501799366334488on_nat(F23,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5946_vebt__memberi__rule,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_memberi(Tia,Xa),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aa_set2974300032405964540o_assn(aTP_Lamm_dk(Na),Sa),Tia),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,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% vebt_memberi_rule
tff(fact_5947_vebt__memberi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine_Imp_refines_o(vEBT_vebt_memberi(Tia,Xa),vEBT_V854960066525838166emberi(T,Tia,Xa)) ).

% vebt_memberi_refines
tff(fact_5948_ln__one__minus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)),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,Xa),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),Xa))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_5949_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xa),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_5950_arcosh__def,axiom,
    ! [Xa: real] : aa_real_real(arcosh_real,Xa) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(powr_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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_5951_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_V854960066525838166emberi(T,Tia,Xa),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_di(T),Tia),Xa)) ).

% vebt_memberi'_rf_abstr
tff(fact_5952_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_5953_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5954_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5955_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = uminus1351360451143612070nteger(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5956_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex(A) = uminus1482373934393186551omplex(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5957_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5958_add_Oinverse__inverse,axiom,
    ! [A: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A)) = A ).

% add.inverse_inverse
tff(fact_5959_add_Oinverse__inverse,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).

% add.inverse_inverse
tff(fact_5960_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : uminus1351360451143612070nteger(uminus1351360451143612070nteger(A)) = A ).

% add.inverse_inverse
tff(fact_5961_add_Oinverse__inverse,axiom,
    ! [A: complex] : uminus1482373934393186551omplex(uminus1482373934393186551omplex(A)) = A ).

% add.inverse_inverse
tff(fact_5962_add_Oinverse__inverse,axiom,
    ! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).

% add.inverse_inverse
tff(fact_5963_verit__minus__simplify_I4_J,axiom,
    ! [B: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B)) = B ).

% verit_minus_simplify(4)
tff(fact_5964_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).

% verit_minus_simplify(4)
tff(fact_5965_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : uminus1351360451143612070nteger(uminus1351360451143612070nteger(B)) = B ).

% verit_minus_simplify(4)
tff(fact_5966_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] : uminus1482373934393186551omplex(uminus1482373934393186551omplex(B)) = B ).

% verit_minus_simplify(4)
tff(fact_5967_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B)) = B ).

% verit_minus_simplify(4)
tff(fact_5968_abs__idempotent,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_idempotent
tff(fact_5969_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_5970_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_5971_abs__idempotent,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_idempotent
tff(fact_5972_arsinh__minus__real,axiom,
    ! [Xa: real] : aa_real_real(arsinh_real,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(arsinh_real,Xa)) ).

% arsinh_minus_real
tff(fact_5973_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_5974_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_5975_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_5976_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_5977_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_5978_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_5979_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = uminus1351360451143612070nteger(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_5980_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_5981_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_5982_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_5983_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_5984_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex(A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_5985_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_5986_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A) )
    <=> ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5987_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5988_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = uminus1351360451143612070nteger(A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5989_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex = uminus1482373934393186551omplex(A) )
    <=> ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5990_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5991_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_5992_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_5993_add_Oinverse__neutral,axiom,
    uminus1351360451143612070nteger(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_5994_add_Oinverse__neutral,axiom,
    uminus1482373934393186551omplex(zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_5995_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_5996_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_5997_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_5998_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_5999_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_6000_compl__le__compl__iff,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xa)),uminus5710092332889474511et_nat(Y))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Y),Xa) ) ).

% compl_le_compl_iff
tff(fact_6001_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_6002_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_6003_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_6004_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_6005_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6006_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6007_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6008_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( uminus1482373934393186551omplex(numera6690914467698888265omplex(M)) = uminus1482373934393186551omplex(numera6690914467698888265omplex(Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6009_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6010_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_6011_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_6012_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) ).

% minus_add_distrib
tff(fact_6013_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : uminus1482373934393186551omplex(plus_plus_complex(A,B)) = plus_plus_complex(uminus1482373934393186551omplex(A),uminus1482373934393186551omplex(B)) ).

% minus_add_distrib
tff(fact_6014_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_6015_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ).

% minus_add_cancel
tff(fact_6016_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).

% minus_add_cancel
tff(fact_6017_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_6018_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : plus_plus_complex(uminus1482373934393186551omplex(A),plus_plus_complex(A,B)) = B ).

% minus_add_cancel
tff(fact_6019_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).

% minus_add_cancel
tff(fact_6020_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_6021_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_6022_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),B)) = B ).

% add_minus_cancel
tff(fact_6023_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : plus_plus_complex(A,plus_plus_complex(uminus1482373934393186551omplex(A),B)) = B ).

% add_minus_cancel
tff(fact_6024_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_6025_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B)) ).

% mult_minus_right
tff(fact_6026_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_right
tff(fact_6027_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : times_3573771949741848930nteger(A,uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(times_3573771949741848930nteger(A,B)) ).

% mult_minus_right
tff(fact_6028_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),uminus1482373934393186551omplex(B)) = uminus1482373934393186551omplex(aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_right
tff(fact_6029_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_right
tff(fact_6030_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(times_times_real(A),B) ).

% minus_mult_minus
tff(fact_6031_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).

% minus_mult_minus
tff(fact_6032_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = times_3573771949741848930nteger(A,B) ).

% minus_mult_minus
tff(fact_6033_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(A)),uminus1482373934393186551omplex(B)) = aa_complex_complex(times_times_complex(A),B) ).

% minus_mult_minus
tff(fact_6034_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) ).

% minus_mult_minus
tff(fact_6035_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B)) ).

% mult_minus_left
tff(fact_6036_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_left
tff(fact_6037_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(A),B) = uminus1351360451143612070nteger(times_3573771949741848930nteger(A,B)) ).

% mult_minus_left
tff(fact_6038_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(A)),B) = uminus1482373934393186551omplex(aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_left
tff(fact_6039_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_left
tff(fact_6040_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(minus_minus_real(B),A) ).

% minus_diff_eq
tff(fact_6041_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% minus_diff_eq
tff(fact_6042_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_6043_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : uminus1482373934393186551omplex(minus_minus_complex(A,B)) = minus_minus_complex(B,A) ).

% minus_diff_eq
tff(fact_6044_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% minus_diff_eq
tff(fact_6045_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).

% div_minus_minus
tff(fact_6046_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_6047_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_6048_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_6049_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_6050_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_6051_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_6052_abs__eq__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_6053_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_6054_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_6055_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_6056_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_6057_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_6058_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_6059_abs__numeral,axiom,
    ! [Na: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,Na) ).

% abs_numeral
tff(fact_6060_abs__numeral,axiom,
    ! [Na: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,Na) ).

% abs_numeral
tff(fact_6061_abs__numeral,axiom,
    ! [Na: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% abs_numeral
tff(fact_6062_abs__numeral,axiom,
    ! [Na: num] : abs_abs_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% abs_numeral
tff(fact_6063_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_minNulli(Tia),aa_VEB6687300313434771872o_assn(aTP_Lamm_dj(T),Tia)) ).

% minNulli_rule
tff(fact_6064_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_6065_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_6066_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_6067_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_6068_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_minus_minus
tff(fact_6069_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_6070_abs__mult__self__eq,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(A)) = times_3573771949741848930nteger(A,A) ).

% abs_mult_self_eq
tff(fact_6071_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(times_times_real(A),A) ).

% abs_mult_self_eq
tff(fact_6072_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% abs_mult_self_eq
tff(fact_6073_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% abs_mult_self_eq
tff(fact_6074_abs__divide,axiom,
    ! [A: complex,B: complex] : abs_abs_complex(divide1717551699836669952omplex(A,B)) = divide1717551699836669952omplex(abs_abs_complex(A),abs_abs_complex(B)) ).

% abs_divide
tff(fact_6075_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_6076_abs__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_divide
tff(fact_6077_of__int__minus,axiom,
    ! [Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,Z)) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_minus
tff(fact_6078_of__int__minus,axiom,
    ! [Z: int] : ring_1_of_int_int(aa_int_int(uminus_uminus_int,Z)) = aa_int_int(uminus_uminus_int,ring_1_of_int_int(Z)) ).

% of_int_minus
tff(fact_6079_of__int__minus,axiom,
    ! [Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z)) = uminus1351360451143612070nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_minus
tff(fact_6080_of__int__minus,axiom,
    ! [Z: int] : ring_17405671764205052669omplex(aa_int_int(uminus_uminus_int,Z)) = uminus1482373934393186551omplex(ring_17405671764205052669omplex(Z)) ).

% of_int_minus
tff(fact_6081_of__int__minus,axiom,
    ! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_minus
tff(fact_6082_abs__minus__cancel,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus_cancel
tff(fact_6083_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_6084_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(uminus1351360451143612070nteger(A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_6085_abs__minus__cancel,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus_cancel
tff(fact_6086_abs__of__nat,axiom,
    ! [Na: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% abs_of_nat
tff(fact_6087_abs__of__nat,axiom,
    ! [Na: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% abs_of_nat
tff(fact_6088_abs__of__nat,axiom,
    ! [Na: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)) = aa_nat_rat(semiri681578069525770553at_rat,Na) ).

% abs_of_nat
tff(fact_6089_abs__of__nat,axiom,
    ! [Na: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) ).

% abs_of_nat
tff(fact_6090_of__int__abs,axiom,
    ! [Xa: int] : ring_1_of_int_int(aa_int_int(abs_abs_int,Xa)) = aa_int_int(abs_abs_int,ring_1_of_int_int(Xa)) ).

% of_int_abs
tff(fact_6091_of__int__abs,axiom,
    ! [Xa: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,Xa)) = abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Xa)) ).

% of_int_abs
tff(fact_6092_of__int__abs,axiom,
    ! [Xa: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,Xa)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Xa)) ).

% of_int_abs
tff(fact_6093_of__int__abs,axiom,
    ! [Xa: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,Xa)) = aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Xa)) ).

% of_int_abs
tff(fact_6094_tanh__minus,axiom,
    ! [Xa: real] : aa_real_real(tanh_real,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(tanh_real,Xa)) ).

% tanh_minus
tff(fact_6095_tanh__minus,axiom,
    ! [Xa: complex] : tanh_complex(uminus1482373934393186551omplex(Xa)) = uminus1482373934393186551omplex(tanh_complex(Xa)) ).

% tanh_minus
tff(fact_6096_tanh__real__abs,axiom,
    ! [Xa: real] : aa_real_real(tanh_real,aa_real_real(abs_abs_real,Xa)) = aa_real_real(abs_abs_real,aa_real_real(tanh_real,Xa)) ).

% tanh_real_abs
tff(fact_6097_vebt__predi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine7594492741263601813on_nat(vEBT_vebt_predi(Tia,Xa),vEBT_VEBT_vebt_predi(T,Tia,Xa)) ).

% vebt_predi_refines
tff(fact_6098_vebt__succi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine7594492741263601813on_nat(vEBT_vebt_succi(Tia,Xa),vEBT_VEBT_vebt_succi(T,Tia,Xa)) ).

% vebt_succi_refines
tff(fact_6099_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_less_eq_nonneg
tff(fact_6100_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_6101_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_6102_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_6103_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_6104_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_6105_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_6106_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_6107_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_le_0_iff_le
tff(fact_6108_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_6109_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_6110_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_6111_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_6112_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_6113_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_6114_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_6115_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_0_iff_less
tff(fact_6116_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_6117_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_6118_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_6119_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_6120_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_6121_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_6122_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_6123_neg__less__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_pos
tff(fact_6124_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_6125_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_6126_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_6127_less__neg__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% less_neg_neg
tff(fact_6128_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_6129_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_6130_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_6131_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_6132_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_6133_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_6134_add_Oright__inverse,axiom,
    ! [A: complex] : plus_plus_complex(A,uminus1482373934393186551omplex(A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_6135_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_6136_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_6137_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_6138_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_6139_ab__left__minus,axiom,
    ! [A: complex] : plus_plus_complex(uminus1482373934393186551omplex(A),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_6140_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_6141_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6142_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6143_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) = uminus1351360451143612070nteger(plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6144_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : plus_plus_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(M)),uminus1482373934393186551omplex(numera6690914467698888265omplex(Na))) = uminus1482373934393186551omplex(plus_plus_complex(numera6690914467698888265omplex(M),numera6690914467698888265omplex(Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6145_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6146_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(minus_minus_real(zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_6147_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_6148_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = uminus1351360451143612070nteger(B) ).

% verit_minus_simplify(3)
tff(fact_6149_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : minus_minus_complex(zero_zero_complex,B) = uminus1482373934393186551omplex(B) ).

% verit_minus_simplify(3)
tff(fact_6150_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_6151_diff__0,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_6152_diff__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_6153_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = uminus1351360451143612070nteger(A) ).

% diff_0
tff(fact_6154_diff__0,axiom,
    ! [A: complex] : minus_minus_complex(zero_zero_complex,A) = uminus1482373934393186551omplex(A) ).

% diff_0
tff(fact_6155_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_6156_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_6157_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_6158_mult__minus1,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(one_one_Code_integer),Z) = uminus1351360451143612070nteger(Z) ).

% mult_minus1
tff(fact_6159_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(one_one_complex)),Z) = uminus1482373934393186551omplex(Z) ).

% mult_minus1
tff(fact_6160_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_6161_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_6162_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_6163_mult__minus1__right,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(Z,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(Z) ).

% mult_minus1_right
tff(fact_6164_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),uminus1482373934393186551omplex(one_one_complex)) = uminus1482373934393186551omplex(Z) ).

% mult_minus1_right
tff(fact_6165_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_6166_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(minus_minus_real(B),A) ).

% uminus_add_conv_diff
tff(fact_6167_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% uminus_add_conv_diff
tff(fact_6168_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_6169_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : plus_plus_complex(uminus1482373934393186551omplex(A),B) = minus_minus_complex(B,A) ).

% uminus_add_conv_diff
tff(fact_6170_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% uminus_add_conv_diff
tff(fact_6171_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).

% diff_minus_eq_add
tff(fact_6172_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).

% diff_minus_eq_add
tff(fact_6173_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,uminus1351360451143612070nteger(B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_6174_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : minus_minus_complex(A,uminus1482373934393186551omplex(B)) = plus_plus_complex(A,B) ).

% diff_minus_eq_add
tff(fact_6175_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).

% diff_minus_eq_add
tff(fact_6176_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_6177_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(A) ).

% div_minus1_right
tff(fact_6178_divide__minus1,axiom,
    ! [Xa: real] : aa_real_real(divide_divide_real(Xa),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xa) ).

% divide_minus1
tff(fact_6179_divide__minus1,axiom,
    ! [Xa: complex] : divide1717551699836669952omplex(Xa,uminus1482373934393186551omplex(one_one_complex)) = uminus1482373934393186551omplex(Xa) ).

% divide_minus1
tff(fact_6180_divide__minus1,axiom,
    ! [Xa: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Xa) ).

% divide_minus1
tff(fact_6181_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_6182_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_6183_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_6184_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_6185_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_6186_abs__le__self__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% abs_le_self_iff
tff(fact_6187_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_6188_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_6189_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_6190_abs__of__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6191_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6192_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6193_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_6194_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_6195_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_6196_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_6197_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_6198_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B) ).

% minus_mod_self1
tff(fact_6199_abs__neg__numeral,axiom,
    ! [Na: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,Na) ).

% abs_neg_numeral
tff(fact_6200_abs__neg__numeral,axiom,
    ! [Na: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,Na) ).

% abs_neg_numeral
tff(fact_6201_abs__neg__numeral,axiom,
    ! [Na: num] : abs_abs_Code_integer(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% abs_neg_numeral
tff(fact_6202_abs__neg__numeral,axiom,
    ! [Na: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,Na) ).

% abs_neg_numeral
tff(fact_6203_inf__compl__bot__left1,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,uminus_uminus_assn(Xa)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)) = bot_bot_assn ).

% inf_compl_bot_left1
tff(fact_6204_inf__compl__bot__left1,axiom,
    ! [Xa: set_real,Y: set_real] : inf_inf_set_real(uminus612125837232591019t_real(Xa),inf_inf_set_real(Xa,Y)) = bot_bot_set_real ).

% inf_compl_bot_left1
tff(fact_6205_inf__compl__bot__left1,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left1
tff(fact_6206_inf__compl__bot__left1,axiom,
    ! [Xa: set_int,Y: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(Xa),inf_inf_set_int(Xa,Y)) = bot_bot_set_int ).

% inf_compl_bot_left1
tff(fact_6207_inf__compl__bot__left2,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,uminus_uminus_assn(Xa)),Y)) = bot_bot_assn ).

% inf_compl_bot_left2
tff(fact_6208_inf__compl__bot__left2,axiom,
    ! [Xa: set_real,Y: set_real] : inf_inf_set_real(Xa,inf_inf_set_real(uminus612125837232591019t_real(Xa),Y)) = bot_bot_set_real ).

% inf_compl_bot_left2
tff(fact_6209_inf__compl__bot__left2,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left2
tff(fact_6210_inf__compl__bot__left2,axiom,
    ! [Xa: set_int,Y: set_int] : inf_inf_set_int(Xa,inf_inf_set_int(uminus1532241313380277803et_int(Xa),Y)) = bot_bot_set_int ).

% inf_compl_bot_left2
tff(fact_6211_inf__compl__bot__right,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),uminus_uminus_assn(Xa))) = bot_bot_assn ).

% inf_compl_bot_right
tff(fact_6212_inf__compl__bot__right,axiom,
    ! [Xa: set_real,Y: set_real] : inf_inf_set_real(Xa,inf_inf_set_real(Y,uminus612125837232591019t_real(Xa))) = bot_bot_set_real ).

% inf_compl_bot_right
tff(fact_6213_inf__compl__bot__right,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),uminus5710092332889474511et_nat(Xa))) = bot_bot_set_nat ).

% inf_compl_bot_right
tff(fact_6214_inf__compl__bot__right,axiom,
    ! [Xa: set_int,Y: set_int] : inf_inf_set_int(Xa,inf_inf_set_int(Y,uminus1532241313380277803et_int(Xa))) = bot_bot_set_int ).

% inf_compl_bot_right
tff(fact_6215_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,uminus_uminus_assn(Xa)),Xa) = bot_bot_assn ).

% boolean_algebra.conj_cancel_left
tff(fact_6216_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_real] : inf_inf_set_real(uminus612125837232591019t_real(Xa),Xa) = bot_bot_set_real ).

% boolean_algebra.conj_cancel_left
tff(fact_6217_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),Xa) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_6218_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(Xa),Xa) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_left
tff(fact_6219_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),uminus_uminus_assn(Xa)) = bot_bot_assn ).

% boolean_algebra.conj_cancel_right
tff(fact_6220_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_real] : inf_inf_set_real(Xa,uminus612125837232591019t_real(Xa)) = bot_bot_set_real ).

% boolean_algebra.conj_cancel_right
tff(fact_6221_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),uminus5710092332889474511et_nat(Xa)) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_6222_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_int] : inf_inf_set_int(Xa,uminus1532241313380277803et_int(Xa)) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_right
tff(fact_6223_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_6224_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_6225_abs__neg__one,axiom,
    abs_abs_Code_integer(uminus1351360451143612070nteger(one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_6226_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_6227_abs__power__minus,axiom,
    ! [A: real,Na: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Na)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ).

% abs_power_minus
tff(fact_6228_abs__power__minus,axiom,
    ! [A: int,Na: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Na)) ).

% abs_power_minus
tff(fact_6229_abs__power__minus,axiom,
    ! [A: code_integer,Na: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Na)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ).

% abs_power_minus
tff(fact_6230_abs__power__minus,axiom,
    ! [A: rat,Na: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Na)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Na)) ).

% abs_power_minus
tff(fact_6231_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [Xa: set_nat,Y: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),uminus5710092332889474511et_nat(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_6232_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [Xa: assn,Y: assn] : uminus_uminus_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,uminus_uminus_assn(Xa)),uminus_uminus_assn(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_6233_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [Xa: set_nat,Y: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(Xa)),uminus5710092332889474511et_nat(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_6234_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [Xa: assn,Y: assn] : uminus_uminus_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,uminus_uminus_assn(Xa)),uminus_uminus_assn(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_6235_real__add__minus__iff,axiom,
    ! [Xa: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( Xa = A ) ) ).

% real_add_minus_iff
tff(fact_6236_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_6237_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_6238_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_6239_add__neg__numeral__special_I7_J,axiom,
    plus_plus_complex(one_one_complex,uminus1482373934393186551omplex(one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_6240_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_6241_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_6242_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_6243_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_6244_add__neg__numeral__special_I8_J,axiom,
    plus_plus_complex(uminus1482373934393186551omplex(one_one_complex),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_6245_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_6246_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6247_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6248_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na)) = uminus1351360451143612070nteger(one_one_Code_integer) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6249_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( uminus1482373934393186551omplex(numera6690914467698888265omplex(Na)) = uminus1482373934393186551omplex(one_one_complex) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6250_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6251_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6252_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6253_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( uminus1351360451143612070nteger(one_one_Code_integer) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6254_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( uminus1482373934393186551omplex(one_one_complex) = uminus1482373934393186551omplex(numera6690914467698888265omplex(Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6255_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6256_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_6257_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_6258_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_6259_diff__numeral__special_I12_J,axiom,
    minus_minus_complex(uminus1482373934393186551omplex(one_one_complex),uminus1482373934393186551omplex(one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_6260_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_6261_left__minus__one__mult__self,axiom,
    ! [Na: nat,A: 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)),Na)),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)),Na)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6262_left__minus__one__mult__self,axiom,
    ! [Na: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6263_left__minus__one__mult__self,axiom,
    ! [Na: nat,A: code_integer] : times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Na),times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Na),A)) = A ).

% left_minus_one_mult_self
tff(fact_6264_left__minus__one__mult__self,axiom,
    ! [Na: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6265_left__minus__one__mult__self,axiom,
    ! [Na: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6266_minus__one__mult__self,axiom,
    ! [Na: 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)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na)) = one_one_real ).

% minus_one_mult_self
tff(fact_6267_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)) = one_one_int ).

% minus_one_mult_self
tff(fact_6268_minus__one__mult__self,axiom,
    ! [Na: nat] : times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Na),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Na)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_6269_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na)),aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na)) = one_one_complex ).

% minus_one_mult_self
tff(fact_6270_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)) = one_one_rat ).

% minus_one_mult_self
tff(fact_6271_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6272_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6273_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6274_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6275_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_6276_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_6277_abs__of__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_6278_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = uminus1351360451143612070nteger(A) ) ) ).

% abs_of_nonpos
tff(fact_6279_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_6280_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_6281_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6282_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6283_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,V)),plus_p5714425477246183910nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(V),W))),Y) ).

% semiring_norm(168)
tff(fact_6284_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] : plus_plus_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(V)),plus_plus_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(W)),Y)) = plus_plus_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W))),Y) ).

% semiring_norm(168)
tff(fact_6285_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6286_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_6287_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_6288_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),Na)) ).

% diff_numeral_simps(2)
tff(fact_6289_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),Na)) ).

% diff_numeral_simps(2)
tff(fact_6290_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),Na)) ).

% diff_numeral_simps(2)
tff(fact_6291_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : minus_minus_complex(numera6690914467698888265omplex(M),uminus1482373934393186551omplex(numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),Na)) ).

% diff_numeral_simps(2)
tff(fact_6292_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),Na)) ).

% diff_numeral_simps(2)
tff(fact_6293_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),Na))) ).

% diff_numeral_simps(3)
tff(fact_6294_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),Na))) ).

% diff_numeral_simps(3)
tff(fact_6295_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),Na))) ).

% diff_numeral_simps(3)
tff(fact_6296_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : minus_minus_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Na)) = uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),Na))) ).

% diff_numeral_simps(3)
tff(fact_6297_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),Na))) ).

% diff_numeral_simps(3)
tff(fact_6298_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6299_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6300_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : times_3573771949741848930nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6301_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(M))),uminus1482373934393186551omplex(numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(aa_num_num(times_times_num(M),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6302_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6303_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6304_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6305_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : times_3573771949741848930nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6306_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(M))),numera6690914467698888265omplex(Na)) = uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6307_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6308_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6309_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6310_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6311_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),uminus1482373934393186551omplex(numera6690914467698888265omplex(Na))) = uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6312_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6313_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: 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)),Y)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6314_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6315_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,V)),times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,W),Y)) = times_3573771949741848930nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(170)
tff(fact_6316_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6317_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6318_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: 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))),Y)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6319_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6320_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,V),times_3573771949741848930nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = times_3573771949741848930nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(171)
tff(fact_6321_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6322_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6323_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: 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))),Y)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_6324_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_6325_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,V)),times_3573771949741848930nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(V),W)),Y) ).

% semiring_norm(172)
tff(fact_6326_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_6327_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_6328_neg__numeral__le__iff,axiom,
    ! [M: num,Na: 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,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M) ) ).

% neg_numeral_le_iff
tff(fact_6329_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M) ) ).

% neg_numeral_le_iff
tff(fact_6330_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M) ) ).

% neg_numeral_le_iff
tff(fact_6331_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M) ) ).

% neg_numeral_le_iff
tff(fact_6332_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M) ) ).

% neg_numeral_less_iff
tff(fact_6333_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M) ) ).

% neg_numeral_less_iff
tff(fact_6334_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M) ) ).

% neg_numeral_less_iff
tff(fact_6335_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M) ) ).

% neg_numeral_less_iff
tff(fact_6336_round__neg__numeral,axiom,
    ! [Na: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% round_neg_numeral
tff(fact_6337_round__neg__numeral,axiom,
    ! [Na: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% round_neg_numeral
tff(fact_6338_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: 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,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6339_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6340_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6341_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6342_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6343_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6344_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))),uminus1351360451143612070nteger(one_one_Code_integer))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6345_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6346_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = 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(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6347_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = divide1717551699836669952omplex(B,uminus1482373934393186551omplex(numera6690914467698888265omplex(W))) )
    <=> $ite(uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),uminus1482373934393186551omplex(numera6690914467698888265omplex(W))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6348_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,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(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6349_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,B = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6350_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( divide1717551699836669952omplex(B,uminus1482373934393186551omplex(numera6690914467698888265omplex(W))) = A )
    <=> $ite(uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),uminus1482373934393186551omplex(numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6351_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6352_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_6353_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_6354_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: 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)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_6355_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_6356_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_6357_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_6358_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: 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)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_6359_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_6360_power2__minus,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6361_power2__minus,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6362_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6363_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6364_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6365_zero__less__power__abs__iff,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na))
    <=> ( ( A != zero_zero_real )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6366_zero__less__power__abs__iff,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Na))
    <=> ( ( A != zero_zero_rat )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6367_zero__less__power__abs__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na))
    <=> ( ( A != zero_zero_int )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6368_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6369_power2__abs,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6370_power2__abs,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6371_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6372_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6373_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6374_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6375_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6376_abs__power2,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6377_artanh__minus__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xa)) ) ) ).

% artanh_minus_real
tff(fact_6378_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6379_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_6380_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6381_add__neg__numeral__special_I9_J,axiom,
    plus_plus_complex(uminus1482373934393186551omplex(one_one_complex),uminus1482373934393186551omplex(one_one_complex)) = uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6382_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6383_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_6384_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_6385_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6386_diff__numeral__special_I10_J,axiom,
    minus_minus_complex(uminus1482373934393186551omplex(one_one_complex),one_one_complex) = uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6387_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6388_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_6389_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_6390_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6391_diff__numeral__special_I11_J,axiom,
    minus_minus_complex(one_one_complex,uminus1482373934393186551omplex(one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6392_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6393_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_6394_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(uminus1351360451143612070nteger(one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_6395_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_6396_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(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_6397_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_6398_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(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_6399_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6400_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6401_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6402_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6403_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6404_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_6405_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_6406_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),one_one_Code_integer) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_6407_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_minus_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(M)),one_one_complex) = uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_6408_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_6409_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_6410_diff__numeral__special_I3_J,axiom,
    ! [Na: 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,Na))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_6411_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_6412_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : minus_minus_complex(one_one_complex,uminus1482373934393186551omplex(numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_6413_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_6414_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6415_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( ring_1_of_int_int(Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6416_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6417_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( ring_17405671764205052669omplex(Y) = aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(Xa))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6418_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6419_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6420_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) = ring_1_of_int_int(Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6421_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa))),Na) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6422_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(Xa))),Na) = ring_17405671764205052669omplex(Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6423_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6424_ceiling__le__neg__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),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,Xa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_6425_ceiling__le__neg__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),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,Xa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_6426_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xa: 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,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xa) ) ).

% neg_numeral_less_ceiling
tff(fact_6427_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xa: 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,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xa) ) ).

% neg_numeral_less_ceiling
tff(fact_6428_ceiling__less__zero,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_6429_ceiling__less__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_6430_zero__le__ceiling,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa) ) ).

% zero_le_ceiling
tff(fact_6431_zero__le__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xa) ) ).

% zero_le_ceiling
tff(fact_6432_power__minus1__even,axiom,
    ! [Na: 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))),Na)) = one_one_real ).

% power_minus1_even
tff(fact_6433_power__minus1__even,axiom,
    ! [Na: nat] : aa_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))),Na)) = one_one_int ).

% power_minus1_even
tff(fact_6434_power__minus1__even,axiom,
    ! [Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(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))),Na)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_6435_power__minus1__even,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_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))),Na)) = one_one_complex ).

% power_minus1_even
tff(fact_6436_power__minus1__even,axiom,
    ! [Na: 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))),Na)) = one_one_rat ).

% power_minus1_even
tff(fact_6437_ceiling__less__neg__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),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,Xa),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_6438_ceiling__less__neg__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),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,Xa),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_6439_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xa: 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,Xa))
    <=> 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)),Xa) ) ).

% neg_numeral_le_ceiling
tff(fact_6440_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xa: 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,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xa) ) ).

% neg_numeral_le_ceiling
tff(fact_6441_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6442_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa))),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6443_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6444_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),ring_1_of_int_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6445_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6446_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6447_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6448_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6449_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6450_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6451_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6452_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6453_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6454_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),ring_1_of_int_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6455_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xa))),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6456_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6457_square__powr__half,axiom,
    ! [Xa: real] : aa_real_real(powr_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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,Xa) ).

% square_powr_half
tff(fact_6458_abs__of__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_6459_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_6460_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = uminus1351360451143612070nteger(A) ) ) ).

% abs_of_neg
tff(fact_6461_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_6462_abs__if__raw,axiom,
    ! [X5: real] :
      aa_real_real(abs_abs_real,X5) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X5),zero_zero_real),aa_real_real(uminus_uminus_real,X5),X5) ).

% abs_if_raw
tff(fact_6463_abs__if__raw,axiom,
    ! [X5: int] :
      aa_int_int(abs_abs_int,X5) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X5),zero_zero_int),aa_int_int(uminus_uminus_int,X5),X5) ).

% abs_if_raw
tff(fact_6464_abs__if__raw,axiom,
    ! [X5: code_integer] :
      abs_abs_Code_integer(X5) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(X5),X5) ).

% abs_if_raw
tff(fact_6465_abs__if__raw,axiom,
    ! [X5: rat] :
      aa_rat_rat(abs_abs_rat,X5) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X5),X5) ).

% abs_if_raw
tff(fact_6466_abs__if,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_if
tff(fact_6467_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).

% abs_if
tff(fact_6468_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A),A) ).

% abs_if
tff(fact_6469_abs__if,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_6470_abs__leI,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).

% abs_leI
tff(fact_6471_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_6472_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B) ) ) ).

% abs_leI
tff(fact_6473_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_6474_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).

% abs_le_D2
tff(fact_6475_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B) ) ).

% abs_le_D2
tff(fact_6476_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% abs_le_D2
tff(fact_6477_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_6478_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_le_iff
tff(fact_6479_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B) ) ) ).

% abs_le_iff
tff(fact_6480_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_le_iff
tff(fact_6481_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_6482_abs__ge__minus__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A)) ).

% abs_ge_minus_self
tff(fact_6483_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_6484_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_minus_self
tff(fact_6485_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_6486_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(uminus_uminus_real,B) = A ) ) ).

% minus_equation_iff
tff(fact_6487_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_6488_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = B )
    <=> ( uminus1351360451143612070nteger(B) = A ) ) ).

% minus_equation_iff
tff(fact_6489_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex(A) = B )
    <=> ( uminus1482373934393186551omplex(B) = A ) ) ).

% minus_equation_iff
tff(fact_6490_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_6491_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_6492_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_6493_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = uminus1351360451143612070nteger(B) )
    <=> ( B = uminus1351360451143612070nteger(A) ) ) ).

% equation_minus_iff
tff(fact_6494_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A = uminus1482373934393186551omplex(B) )
    <=> ( B = uminus1482373934393186551omplex(A) ) ) ).

% equation_minus_iff
tff(fact_6495_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_6496_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_6497_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_6498_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( uminus1351360451143612070nteger(A) = uminus1351360451143612070nteger(B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_6499_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_6500_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_less_iff
tff(fact_6501_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_6502_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),B) ) ) ).

% abs_less_iff
tff(fact_6503_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_less_iff
tff(fact_6504_abs__minus__le__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_6505_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_6506_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_6507_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_6508_abs__real__def,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_real_def
tff(fact_6509_abs__ge__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(abs_abs_real,A)) ).

% abs_ge_self
tff(fact_6510_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A)) ).

% abs_ge_self
tff(fact_6511_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_self
tff(fact_6512_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_6513_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% abs_le_D1
tff(fact_6514_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_6515_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_6516_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_6517_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(times_3573771949741848930nteger(A,B)) = times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_6518_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_6519_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_6520_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_6521_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_6522_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)) ).

% abs_minus_commute
tff(fact_6523_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)) ).

% abs_minus_commute
tff(fact_6524_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_6525_power__abs,axiom,
    ! [A: int,Na: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Na)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na) ).

% power_abs
tff(fact_6526_power__abs,axiom,
    ! [A: real,Na: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na) ).

% power_abs
tff(fact_6527_power__abs,axiom,
    ! [A: code_integer,Na: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na) ).

% power_abs
tff(fact_6528_power__abs,axiom,
    ! [A: rat,Na: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Na)) = aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Na) ).

% power_abs
tff(fact_6529_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_6530_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A)) ) ).

% le_imp_neg_le
tff(fact_6531_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_6532_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_6533_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_6534_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),A) ) ).

% minus_le_iff
tff(fact_6535_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_le_iff
tff(fact_6536_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_6537_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_6538_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),uminus1351360451143612070nteger(B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),uminus1351360451143612070nteger(A)) ) ).

% le_minus_iff
tff(fact_6539_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_6540_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_6541_compl__le__swap2,axiom,
    ! [Y: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),Xa)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xa)),Y) ) ).

% compl_le_swap2
tff(fact_6542_compl__le__swap1,axiom,
    ! [Y: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(Xa))
     => aa_set_nat_o(ord_less_eq_set_nat(Xa),uminus5710092332889474511et_nat(Y)) ) ).

% compl_le_swap1
tff(fact_6543_compl__mono,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(Xa)) ) ).

% compl_mono
tff(fact_6544_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_6545_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_6546_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),A) ) ).

% minus_less_iff
tff(fact_6547_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_less_iff
tff(fact_6548_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_6549_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_6550_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),uminus1351360451143612070nteger(B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),uminus1351360451143612070nteger(A)) ) ).

% less_minus_iff
tff(fact_6551_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_6552_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_6553_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_6554_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A)) ) ).

% verit_negate_coefficient(2)
tff(fact_6555_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_6556_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : aa_num_real(numeral_numeral_real,M) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% numeral_neq_neg_numeral
tff(fact_6557_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% numeral_neq_neg_numeral
tff(fact_6558_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : aa_num_Code_integer(numera6620942414471956472nteger,M) != uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% numeral_neq_neg_numeral
tff(fact_6559_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : numera6690914467698888265omplex(M) != uminus1482373934393186551omplex(numera6690914467698888265omplex(Na)) ).

% numeral_neq_neg_numeral
tff(fact_6560_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : aa_num_rat(numeral_numeral_rat,M) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% numeral_neq_neg_numeral
tff(fact_6561_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) != aa_num_real(numeral_numeral_real,Na) ).

% neg_numeral_neq_numeral
tff(fact_6562_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,Na) ).

% neg_numeral_neq_numeral
tff(fact_6563_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)) != aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% neg_numeral_neq_numeral
tff(fact_6564_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : uminus1482373934393186551omplex(numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(Na) ).

% neg_numeral_neq_numeral
tff(fact_6565_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) != aa_num_rat(numeral_numeral_rat,Na) ).

% neg_numeral_neq_numeral
tff(fact_6566_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_6567_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_6568_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(B),uminus1351360451143612070nteger(A)) ).

% is_num_normalize(8)
tff(fact_6569_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : uminus1482373934393186551omplex(plus_plus_complex(A,B)) = plus_plus_complex(uminus1482373934393186551omplex(B),uminus1482373934393186551omplex(A)) ).

% is_num_normalize(8)
tff(fact_6570_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_6571_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_6572_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_6573_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(B),uminus1351360451143612070nteger(A)) ).

% add.inverse_distrib_swap
tff(fact_6574_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : uminus1482373934393186551omplex(plus_plus_complex(A,B)) = plus_plus_complex(uminus1482373934393186551omplex(B),uminus1482373934393186551omplex(A)) ).

% add.inverse_distrib_swap
tff(fact_6575_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_6576_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_6577_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_6578_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( uminus1351360451143612070nteger(A2) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K),uminus1351360451143612070nteger(A)) ) ) ).

% group_cancel.neg1
tff(fact_6579_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = plus_plus_complex(K,A) )
     => ( uminus1482373934393186551omplex(A2) = plus_plus_complex(uminus1482373934393186551omplex(K),uminus1482373934393186551omplex(A)) ) ) ).

% group_cancel.neg1
tff(fact_6580_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_6581_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_6582_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_6583_one__neq__neg__one,axiom,
    one_one_Code_integer != uminus1351360451143612070nteger(one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_6584_one__neq__neg__one,axiom,
    one_one_complex != uminus1482373934393186551omplex(one_one_complex) ).

% one_neq_neg_one
tff(fact_6585_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_6586_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_6587_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_6588_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(A),B) = times_3573771949741848930nteger(A,uminus1351360451143612070nteger(B)) ).

% minus_mult_commute
tff(fact_6589_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(A)),B) = aa_complex_complex(times_times_complex(A),uminus1482373934393186551omplex(B)) ).

% minus_mult_commute
tff(fact_6590_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_6591_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),A) = aa_real_real(times_times_real(B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_6592_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_6593_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger(A,A) = times_3573771949741848930nteger(B,B) )
    <=> ( ( A = B )
        | ( A = uminus1351360451143612070nteger(B) ) ) ) ).

% square_eq_iff
tff(fact_6594_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B),B) )
    <=> ( ( A = B )
        | ( A = uminus1482373934393186551omplex(B) ) ) ) ).

% square_eq_iff
tff(fact_6595_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_6596_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_6597_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_6598_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(B),A) = minus_8373710615458151222nteger(uminus1351360451143612070nteger(A),B) ).

% minus_diff_commute
tff(fact_6599_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : minus_minus_complex(uminus1482373934393186551omplex(B),A) = minus_minus_complex(uminus1482373934393186551omplex(A),B) ).

% minus_diff_commute
tff(fact_6600_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_6601_minus__diff__minus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B)) ).

% minus_diff_minus
tff(fact_6602_minus__diff__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ).

% minus_diff_minus
tff(fact_6603_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(minus_8373710615458151222nteger(A,B)) ).

% minus_diff_minus
tff(fact_6604_minus__diff__minus,axiom,
    ! [A: complex,B: complex] : minus_minus_complex(uminus1482373934393186551omplex(A),uminus1482373934393186551omplex(B)) = uminus1482373934393186551omplex(minus_minus_complex(A,B)) ).

% minus_diff_minus
tff(fact_6605_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ).

% minus_diff_minus
tff(fact_6606_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_6607_minus__divide__right,axiom,
    ! [A: complex,B: complex] : uminus1482373934393186551omplex(divide1717551699836669952omplex(A,B)) = divide1717551699836669952omplex(A,uminus1482373934393186551omplex(B)) ).

% minus_divide_right
tff(fact_6608_minus__divide__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_divide_right
tff(fact_6609_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(A),B) ).

% minus_divide_divide
tff(fact_6610_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : divide1717551699836669952omplex(uminus1482373934393186551omplex(A),uminus1482373934393186551omplex(B)) = divide1717551699836669952omplex(A,B) ).

% minus_divide_divide
tff(fact_6611_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ).

% minus_divide_divide
tff(fact_6612_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_6613_minus__divide__left,axiom,
    ! [A: complex,B: complex] : uminus1482373934393186551omplex(divide1717551699836669952omplex(A,B)) = divide1717551699836669952omplex(uminus1482373934393186551omplex(A),B) ).

% minus_divide_left
tff(fact_6614_minus__divide__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_divide_left
tff(fact_6615_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_6616_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,uminus1351360451143612070nteger(B)) = divide6298287555418463151nteger(uminus1351360451143612070nteger(A),B) ).

% div_minus_right
tff(fact_6617_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_6618_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(uminus1351360451143612070nteger(modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B) ).

% mod_minus_eq
tff(fact_6619_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: int,B: int,A3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),B) )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A3)),B) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_6620_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A3,B) )
     => ( modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A3),B) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_6621_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_6622_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B)) ).

% mod_minus_right
tff(fact_6623_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_6624_of__int__neg__numeral,axiom,
    ! [K: num] : 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_6625_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))) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,K)) ).

% of_int_neg_numeral
tff(fact_6626_of__int__neg__numeral,axiom,
    ! [K: num] : ring_17405671764205052669omplex(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus1482373934393186551omplex(numera6690914467698888265omplex(K)) ).

% of_int_neg_numeral
tff(fact_6627_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_6628_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).

% abs_ge_zero
tff(fact_6629_abs__ge__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A)) ).

% abs_ge_zero
tff(fact_6630_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_zero
tff(fact_6631_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_6632_abs__not__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_6633_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_6634_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_6635_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_6636_abs__of__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_6637_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_6638_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_6639_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_6640_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_6641_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_6642_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_6643_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_6644_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),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,A)),aa_real_real(abs_abs_real,B))),aa_real_real(times_times_real(C),D)) ) ) ).

% abs_mult_less
tff(fact_6645_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D)) ) ) ).

% abs_mult_less
tff(fact_6646_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),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,A)),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_6647_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),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,times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),times_3573771949741848930nteger(C,D)) ) ) ).

% abs_mult_less
tff(fact_6648_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_6649_abs__triangle__ineq2__sym,axiom,
    ! [A: 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,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6650_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6651_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6652_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq3
tff(fact_6653_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))) ).

% abs_triangle_ineq3
tff(fact_6654_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq3
tff(fact_6655_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq3
tff(fact_6656_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq2
tff(fact_6657_abs__triangle__ineq2,axiom,
    ! [A: 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,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))) ).

% abs_triangle_ineq2
tff(fact_6658_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq2
tff(fact_6659_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq2
tff(fact_6660_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_6661_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_6662_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% zero_neq_neg_numeral
tff(fact_6663_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% zero_neq_neg_numeral
tff(fact_6664_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_z3403309356797280102nteger != uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% zero_neq_neg_numeral
tff(fact_6665_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_complex != uminus1482373934393186551omplex(numera6690914467698888265omplex(Na)) ).

% zero_neq_neg_numeral
tff(fact_6666_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_neq_neg_numeral
tff(fact_6667_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_6668_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_6669_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_6670_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_6671_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: 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,M))),aa_num_real(numeral_numeral_real,Na)) ).

% neg_numeral_le_numeral
tff(fact_6672_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% neg_numeral_le_numeral
tff(fact_6673_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Na)) ).

% neg_numeral_le_numeral
tff(fact_6674_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) ).

% neg_numeral_le_numeral
tff(fact_6675_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_6676_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_6677_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_6678_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_6679_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Na)) ).

% neg_numeral_less_numeral
tff(fact_6680_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) ).

% neg_numeral_less_numeral
tff(fact_6681_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% neg_numeral_less_numeral
tff(fact_6682_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Na)) ).

% neg_numeral_less_numeral
tff(fact_6683_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_6684_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_6685_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = uminus1351360451143612070nteger(A) ) ) ).

% add_eq_0_iff
tff(fact_6686_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex(A,B) = zero_zero_complex )
    <=> ( B = uminus1482373934393186551omplex(A) ) ) ).

% add_eq_0_iff
tff(fact_6687_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_6688_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_6689_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_6690_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_6691_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : plus_plus_complex(uminus1482373934393186551omplex(A),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_6692_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_6693_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_6694_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_6695_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( uminus1351360451143612070nteger(A) = B ) ) ).

% add.inverse_unique
tff(fact_6696_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex(A,B) = zero_zero_complex )
     => ( uminus1482373934393186551omplex(A) = B ) ) ).

% add.inverse_unique
tff(fact_6697_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_6698_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6699_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6700_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = uminus1351360451143612070nteger(B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6701_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = uminus1482373934393186551omplex(B) )
    <=> ( plus_plus_complex(A,B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6702_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6703_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6704_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6705_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6706_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex(A) = B )
    <=> ( plus_plus_complex(A,B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6707_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6708_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_6709_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_6710_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != uminus1351360451143612070nteger(one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_6711_zero__neq__neg__one,axiom,
    zero_zero_complex != uminus1482373934393186551omplex(one_one_complex) ).

% zero_neq_neg_one
tff(fact_6712_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_6713_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_6714_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_6715_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_6716_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_6717_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_6718_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_6719_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_6720_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_6721_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_6722_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_6723_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_6724_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_6725_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_6726_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_6727_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_6728_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_6729_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% one_neq_neg_numeral
tff(fact_6730_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% one_neq_neg_numeral
tff(fact_6731_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_Code_integer != uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% one_neq_neg_numeral
tff(fact_6732_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_complex != uminus1482373934393186551omplex(numera6690914467698888265omplex(Na)) ).

% one_neq_neg_numeral
tff(fact_6733_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% one_neq_neg_numeral
tff(fact_6734_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,Na) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_6735_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,Na) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_6736_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_Code_integer(numera6620942414471956472nteger,Na) != uminus1351360451143612070nteger(one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_6737_numeral__neq__neg__one,axiom,
    ! [Na: num] : numera6690914467698888265omplex(Na) != uminus1482373934393186551omplex(one_one_complex) ).

% numeral_neq_neg_one
tff(fact_6738_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,Na) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_6739_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(times_times_real(Xa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_6740_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: 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,Xa)) = aa_int_int(aa_int_fun_int_int(times_times_int,Xa),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_6741_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: code_integer] : times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,W),uminus1351360451143612070nteger(Xa)) = times_3573771949741848930nteger(Xa,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,W))) ).

% numeral_times_minus_swap
tff(fact_6742_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),uminus1482373934393186551omplex(Xa)) = aa_complex_complex(times_times_complex(Xa),uminus1482373934393186551omplex(numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_6743_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,Xa)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_6744_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6745_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( divide1717551699836669952omplex(uminus1482373934393186551omplex(A),uminus1482373934393186551omplex(B)) = divide1717551699836669952omplex(A,B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6746_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6747_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6748_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( uminus1482373934393186551omplex(divide1717551699836669952omplex(A,B)) = divide1717551699836669952omplex(A,uminus1482373934393186551omplex(B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6749_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6750_square__eq__1__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(times_times_real(Xa),Xa) = one_one_real )
    <=> ( ( Xa = one_one_real )
        | ( Xa = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_6751_square__eq__1__iff,axiom,
    ! [Xa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa) = one_one_int )
    <=> ( ( Xa = one_one_int )
        | ( Xa = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_6752_square__eq__1__iff,axiom,
    ! [Xa: code_integer] :
      ( ( times_3573771949741848930nteger(Xa,Xa) = one_one_Code_integer )
    <=> ( ( Xa = one_one_Code_integer )
        | ( Xa = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_6753_square__eq__1__iff,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_complex(times_times_complex(Xa),Xa) = one_one_complex )
    <=> ( ( Xa = one_one_complex )
        | ( Xa = uminus1482373934393186551omplex(one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_6754_square__eq__1__iff,axiom,
    ! [Xa: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa) = one_one_rat )
    <=> ( ( Xa = one_one_rat )
        | ( Xa = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_6755_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(minus_minus_real(A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6756_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6757_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_6758_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2 = plus_plus_complex(K,B) )
     => ( minus_minus_complex(A,B2) = plus_plus_complex(uminus1482373934393186551omplex(K),minus_minus_complex(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_6759_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6760_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_6761_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_6762_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(B)) ).

% diff_conv_add_uminus
tff(fact_6763_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : minus_minus_complex(A,B) = plus_plus_complex(A,uminus1482373934393186551omplex(B)) ).

% diff_conv_add_uminus
tff(fact_6764_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_6765_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6766_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6767_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6768_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : minus_minus_complex(A,B) = plus_plus_complex(A,uminus1482373934393186551omplex(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6769_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6770_diff__eq,axiom,
    ! [Xa: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,Xa),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),uminus_uminus_assn(Y)) ).

% diff_eq
tff(fact_6771_diff__eq,axiom,
    ! [Xa: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Xa),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),uminus5710092332889474511et_nat(Y)) ).

% diff_eq
tff(fact_6772_inf__cancel__left1,axiom,
    ! [Xa: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),A)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,uminus_uminus_assn(Xa)),B)) = bot_bot_assn ).

% inf_cancel_left1
tff(fact_6773_inf__cancel__left1,axiom,
    ! [Xa: set_real,A: set_real,B: set_real] : inf_inf_set_real(inf_inf_set_real(Xa,A),inf_inf_set_real(uminus612125837232591019t_real(Xa),B)) = bot_bot_set_real ).

% inf_cancel_left1
tff(fact_6774_inf__cancel__left1,axiom,
    ! [Xa: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),B)) = bot_bot_set_nat ).

% inf_cancel_left1
tff(fact_6775_inf__cancel__left1,axiom,
    ! [Xa: set_int,A: set_int,B: set_int] : inf_inf_set_int(inf_inf_set_int(Xa,A),inf_inf_set_int(uminus1532241313380277803et_int(Xa),B)) = bot_bot_set_int ).

% inf_cancel_left1
tff(fact_6776_inf__cancel__left2,axiom,
    ! [Xa: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,uminus_uminus_assn(Xa)),A)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),B)) = bot_bot_assn ).

% inf_cancel_left2
tff(fact_6777_inf__cancel__left2,axiom,
    ! [Xa: set_real,A: set_real,B: set_real] : inf_inf_set_real(inf_inf_set_real(uminus612125837232591019t_real(Xa),A),inf_inf_set_real(Xa,B)) = bot_bot_set_real ).

% inf_cancel_left2
tff(fact_6778_inf__cancel__left2,axiom,
    ! [Xa: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),B)) = bot_bot_set_nat ).

% inf_cancel_left2
tff(fact_6779_inf__cancel__left2,axiom,
    ! [Xa: set_int,A: set_int,B: set_int] : inf_inf_set_int(inf_inf_set_int(uminus1532241313380277803et_int(Xa),A),inf_inf_set_int(Xa,B)) = bot_bot_set_int ).

% inf_cancel_left2
tff(fact_6780_real__minus__mult__self__le,axiom,
    ! [U: real,Xa: 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(Xa),Xa)) ).

% real_minus_mult_self_le
tff(fact_6781_minus__real__def,axiom,
    ! [Xa: real,Y: real] : aa_real_real(minus_minus_real(Xa),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_6782_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6783_of__int__of__nat,axiom,
    ! [K: int] :
      ring_1_of_int_int(K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6784_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6785_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6786_of__int__of__nat,axiom,
    ! [K: int] :
      ring_17405671764205052669omplex(K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),uminus1482373934393186551omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6787_dense__eq0__I,axiom,
    ! [Xa: 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,Xa)),E2) )
     => ( Xa = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_6788_dense__eq0__I,axiom,
    ! [Xa: 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,Xa)),E2) )
     => ( Xa = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_6789_abs__mult__pos,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( times_3573771949741848930nteger(abs_abs_Code_integer(Y),Xa) = abs_abs_Code_integer(times_3573771949741848930nteger(Y,Xa)) ) ) ).

% abs_mult_pos
tff(fact_6790_abs__mult__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(times_times_real(aa_real_real(abs_abs_real,Y)),Xa) = aa_real_real(abs_abs_real,aa_real_real(times_times_real(Y),Xa)) ) ) ).

% abs_mult_pos
tff(fact_6791_abs__mult__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),Xa) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xa)) ) ) ).

% abs_mult_pos
tff(fact_6792_abs__mult__pos,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),Xa) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xa)) ) ) ).

% abs_mult_pos
tff(fact_6793_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(times_3573771949741848930nteger(A,B)) = times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_6794_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_6795_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_6796_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_6797_abs__div__pos,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,Xa)),Y) = aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Xa),Y)) ) ) ).

% abs_div_pos
tff(fact_6798_abs__div__pos,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,Xa)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Y)) ) ) ).

% abs_div_pos
tff(fact_6799_zero__le__power__abs,axiom,
    ! [A: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na)) ).

% zero_le_power_abs
tff(fact_6800_zero__le__power__abs,axiom,
    ! [A: code_integer,Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na)) ).

% zero_le_power_abs
tff(fact_6801_zero__le__power__abs,axiom,
    ! [A: rat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Na)) ).

% zero_le_power_abs
tff(fact_6802_zero__le__power__abs,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na)) ).

% zero_le_power_abs
tff(fact_6803_abs__diff__le__iff,axiom,
    ! [Xa: code_integer,A: code_integer,R: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xa,A))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R)),Xa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),plus_p5714425477246183910nteger(A,R)) ) ) ).

% abs_diff_le_iff
tff(fact_6804_abs__diff__le__iff,axiom,
    ! [Xa: real,A: real,R: 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(Xa),A))),R)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),R)),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_6805_abs__diff__le__iff,axiom,
    ! [Xa: rat,A: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),A))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_6806_abs__diff__le__iff,axiom,
    ! [Xa: int,A: int,R: 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,Xa),A))),R)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),Xa)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_6807_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_6808_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_6809_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_6810_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_6811_abs__diff__triangle__ineq,axiom,
    ! [A: 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(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_6812_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6813_abs__diff__triangle__ineq,axiom,
    ! [A: 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(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6814_abs__diff__triangle__ineq,axiom,
    ! [A: 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,A),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,A),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_6815_abs__diff__less__iff,axiom,
    ! [Xa: real,A: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xa),A))),R)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),R)),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_6816_abs__diff__less__iff,axiom,
    ! [Xa: rat,A: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),A))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_6817_abs__diff__less__iff,axiom,
    ! [Xa: int,A: int,R: 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,Xa),A))),R)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),Xa)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_6818_abs__diff__less__iff,axiom,
    ! [Xa: code_integer,A: code_integer,R: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xa,A))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R)),Xa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),plus_p5714425477246183910nteger(A,R)) ) ) ).

% abs_diff_less_iff
tff(fact_6819_not__zero__le__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_le_neg_numeral
tff(fact_6820_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% not_zero_le_neg_numeral
tff(fact_6821_not__zero__le__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_le_neg_numeral
tff(fact_6822_not__zero__le__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_le_neg_numeral
tff(fact_6823_neg__numeral__le__zero,axiom,
    ! [Na: 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,Na))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_6824_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_6825_neg__numeral__le__zero,axiom,
    ! [Na: 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,Na))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_6826_neg__numeral__le__zero,axiom,
    ! [Na: 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,Na))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_6827_not__zero__less__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_less_neg_numeral
tff(fact_6828_not__zero__less__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_less_neg_numeral
tff(fact_6829_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% not_zero_less_neg_numeral
tff(fact_6830_not__zero__less__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_less_neg_numeral
tff(fact_6831_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_6832_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_6833_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_6834_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_6835_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_6836_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_6837_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_6838_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_6839_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_6840_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_6841_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_6842_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_6843_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_6844_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_6845_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_6846_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_6847_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_6848_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_6849_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_6850_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_6851_not__one__le__neg__numeral,axiom,
    ! [M: 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,M))) ).

% not_one_le_neg_numeral
tff(fact_6852_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_one_le_neg_numeral
tff(fact_6853_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))) ).

% not_one_le_neg_numeral
tff(fact_6854_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_le_neg_numeral
tff(fact_6855_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_6856_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_6857_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_6858_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_6859_neg__numeral__le__neg__one,axiom,
    ! [M: 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,M))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_6860_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_6861_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_6862_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_6863_neg__one__le__numeral,axiom,
    ! [M: 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,M)) ).

% neg_one_le_numeral
tff(fact_6864_neg__one__le__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M)) ).

% neg_one_le_numeral
tff(fact_6865_neg__one__le__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M)) ).

% neg_one_le_numeral
tff(fact_6866_neg__one__le__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_le_numeral
tff(fact_6867_neg__numeral__le__one,axiom,
    ! [M: 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,M))),one_one_real) ).

% neg_numeral_le_one
tff(fact_6868_neg__numeral__le__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_6869_neg__numeral__le__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_6870_neg__numeral__le__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_le_one
tff(fact_6871_not__neg__one__less__neg__numeral,axiom,
    ! [M: 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,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_6872_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_6873_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_6874_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_6875_not__one__less__neg__numeral,axiom,
    ! [M: 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,M))) ).

% not_one_less_neg_numeral
tff(fact_6876_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_less_neg_numeral
tff(fact_6877_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_one_less_neg_numeral
tff(fact_6878_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))) ).

% not_one_less_neg_numeral
tff(fact_6879_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_6880_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_6881_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_6882_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_6883_neg__one__less__numeral,axiom,
    ! [M: 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,M)) ).

% neg_one_less_numeral
tff(fact_6884_neg__one__less__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_less_numeral
tff(fact_6885_neg__one__less__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M)) ).

% neg_one_less_numeral
tff(fact_6886_neg__one__less__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M)) ).

% neg_one_less_numeral
tff(fact_6887_neg__numeral__less__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) ).

% neg_numeral_less_one
tff(fact_6888_neg__numeral__less__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_less_one
tff(fact_6889_neg__numeral__less__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_6890_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_6891_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_6892_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_6893_uminus__numeral__One,axiom,
    uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,one)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_6894_uminus__numeral__One,axiom,
    uminus1482373934393186551omplex(numera6690914467698888265omplex(one)) = uminus1482373934393186551omplex(one_one_complex) ).

% uminus_numeral_One
tff(fact_6895_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_6896_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_6897_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_6898_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,one)),B) = uminus1351360451143612070nteger(B) ).

% mult_1s_ring_1(1)
tff(fact_6899_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(one))),B) = uminus1482373934393186551omplex(B) ).

% mult_1s_ring_1(1)
tff(fact_6900_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_6901_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_6902_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_6903_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : times_3573771949741848930nteger(B,uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,one))) = uminus1351360451143612070nteger(B) ).

% mult_1s_ring_1(2)
tff(fact_6904_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(B),uminus1482373934393186551omplex(numera6690914467698888265omplex(one))) = uminus1482373934393186551omplex(B) ).

% mult_1s_ring_1(2)
tff(fact_6905_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_6906_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6907_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex(A,B) = uminus1482373934393186551omplex(one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( A = uminus1482373934393186551omplex(B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6908_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6909_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = 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(A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_6910_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = uminus1482373934393186551omplex(divide1717551699836669952omplex(B,C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = uminus1482373934393186551omplex(B),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_6911_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_6912_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(times_times_real(A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_6913_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( uminus1482373934393186551omplex(divide1717551699836669952omplex(B,C)) = A )
    <=> $ite(C != zero_zero_complex,uminus1482373934393186551omplex(B) = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_6914_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) = A )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_6915_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(times_times_real(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6916_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex(divide1717551699836669952omplex(A,B)) = C )
      <=> ( uminus1482373934393186551omplex(A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6917_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6918_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) )
      <=> ( aa_real_real(times_times_real(C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6919_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = uminus1482373934393186551omplex(divide1717551699836669952omplex(A,B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = uminus1482373934393186551omplex(A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6920_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6921_power__minus,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Na) = 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)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ).

% power_minus
tff(fact_6922_power__minus,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)),aa_nat_int(power_power_int(A),Na)) ).

% power_minus
tff(fact_6923_power__minus,axiom,
    ! [A: code_integer,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Na) = times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Na),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ).

% power_minus
tff(fact_6924_power__minus,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(A)),Na) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na)),aa_nat_complex(power_power_complex(A),Na)) ).

% power_minus
tff(fact_6925_power__minus,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Na) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)),aa_nat_rat(power_power_rat(A),Na)) ).

% power_minus
tff(fact_6926_power__minus__Bit0,axiom,
    ! [Xa: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6927_power__minus__Bit0,axiom,
    ! [Xa: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6928_power__minus__Bit0,axiom,
    ! [Xa: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6929_power__minus__Bit0,axiom,
    ! [Xa: complex,K: num] : aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6930_power__minus__Bit0,axiom,
    ! [Xa: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6931_inf__shunt,axiom,
    ! [Xa: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y) = bot_bot_assn )
    <=> ord_less_eq_assn(Xa,uminus_uminus_assn(Y)) ) ).

% inf_shunt
tff(fact_6932_inf__shunt,axiom,
    ! [Xa: set_real,Y: set_real] :
      ( ( inf_inf_set_real(Xa,Y) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xa),uminus612125837232591019t_real(Y)) ) ).

% inf_shunt
tff(fact_6933_inf__shunt,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( inf_inf_set_int(Xa,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),uminus1532241313380277803et_int(Y)) ) ).

% inf_shunt
tff(fact_6934_inf__shunt,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xa),uminus5710092332889474511et_nat(Y)) ) ).

% inf_shunt
tff(fact_6935_power__minus__Bit1,axiom,
    ! [Xa: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xa)),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6936_power__minus__Bit1,axiom,
    ! [Xa: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6937_power__minus__Bit1,axiom,
    ! [Xa: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus1351360451143612070nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6938_power__minus__Bit1,axiom,
    ! [Xa: complex,K: num] : aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus1482373934393186551omplex(aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6939_power__minus__Bit1,axiom,
    ! [Xa: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6940_sup__neg__inf,axiom,
    ! [P3: assn,Q2: assn,R: assn] :
      ( ord_less_eq_assn(P3,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Q2),R))
    <=> ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P3),uminus_uminus_assn(Q2)),R) ) ).

% sup_neg_inf
tff(fact_6941_sup__neg__inf,axiom,
    ! [P3: set_nat,Q2: set_nat,R: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(P3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q2),R))
    <=> aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,P3),uminus5710092332889474511et_nat(Q2))),R) ) ).

% sup_neg_inf
tff(fact_6942_shunt2,axiom,
    ! [Xa: assn,Y: assn,Z: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),uminus_uminus_assn(Y)),Z)
    <=> ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ) ).

% shunt2
tff(fact_6943_shunt2,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),uminus5710092332889474511et_nat(Y))),Z)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ) ).

% shunt2
tff(fact_6944_shunt1,axiom,
    ! [Xa: assn,Y: assn,Z: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),Y),Z)
    <=> ord_less_eq_assn(Xa,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,uminus_uminus_assn(Y)),Z)) ) ).

% shunt1
tff(fact_6945_shunt1,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Y)),Z)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(Y)),Z)) ) ).

% shunt1
tff(fact_6946_lemma__interval__lt,axiom,
    ! [A: real,Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xa),Y4))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_6947_powr__minus__divide,axiom,
    ! [Xa: real,A: real] : aa_real_real(powr_real(Xa),aa_real_real(uminus_uminus_real,A)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(powr_real(Xa),A)) ).

% powr_minus_divide
tff(fact_6948_real__add__less__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xa)) ) ).

% real_add_less_0_iff
tff(fact_6949_real__0__less__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xa)),Y) ) ).

% real_0_less_add_iff
tff(fact_6950_real__0__le__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xa)),Y) ) ).

% real_0_le_add_iff
tff(fact_6951_real__add__le__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,Xa)) ) ).

% real_add_le_0_iff
tff(fact_6952_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(A),aa_real_real(powr_real(B),C)) = aa_real_real(times_times_real(A),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_6953_tanh__real__gt__neg1,axiom,
    ! [Xa: 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,Xa)) ).

% tanh_real_gt_neg1
tff(fact_6954_abs__add__one__gt__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xa))) ).

% abs_add_one_gt_zero
tff(fact_6955_abs__add__one__gt__zero,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,Xa))) ).

% abs_add_one_gt_zero
tff(fact_6956_abs__add__one__gt__zero,axiom,
    ! [Xa: 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,Xa))) ).

% abs_add_one_gt_zero
tff(fact_6957_abs__add__one__gt__zero,axiom,
    ! [Xa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xa))) ).

% abs_add_one_gt_zero
tff(fact_6958_of__int__leD,axiom,
    ! [Na: int,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xa) ) ) ).

% of_int_leD
tff(fact_6959_of__int__leD,axiom,
    ! [Na: int,Xa: 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,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% of_int_leD
tff(fact_6960_of__int__leD,axiom,
    ! [Na: int,Xa: 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,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ) ).

% of_int_leD
tff(fact_6961_of__int__leD,axiom,
    ! [Na: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,ring_1_of_int_int(Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xa) ) ) ).

% of_int_leD
tff(fact_6962_of__int__lessD,axiom,
    ! [Na: int,Xa: 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,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% of_int_lessD
tff(fact_6963_of__int__lessD,axiom,
    ! [Na: int,Xa: 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,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xa) ) ) ).

% of_int_lessD
tff(fact_6964_of__int__lessD,axiom,
    ! [Na: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,ring_1_of_int_int(Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xa) ) ) ).

% of_int_lessD
tff(fact_6965_of__int__lessD,axiom,
    ! [Na: int,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xa) ) ) ).

% of_int_lessD
tff(fact_6966_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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(A),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(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_6967_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_6968_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: 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))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_6969_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_6970_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(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(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6971_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6972_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6973_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6974_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(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(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6975_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6976_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6977_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6978_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_6979_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) = divide1717551699836669952omplex(B,C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(W))),C) = B,uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6980_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(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,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_6981_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_6982_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( divide1717551699836669952omplex(B,C) = uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(numera6690914467698888265omplex(W))),C),uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6983_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,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_6984_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6985_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      plus_plus_complex(uminus1482373934393186551omplex(divide1717551699836669952omplex(A,Z)),B) = $ite(Z = zero_zero_complex,B,divide1717551699836669952omplex(plus_plus_complex(uminus1482373934393186551omplex(A),aa_complex_complex(times_times_complex(B),Z)),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6986_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6987_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xa),Z))),Y) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Xa)),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6988_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( plus_plus_complex(uminus1482373934393186551omplex(divide1717551699836669952omplex(Xa,Z)),Y) = divide1717551699836669952omplex(plus_plus_complex(uminus1482373934393186551omplex(Xa),aa_complex_complex(times_times_complex(Y),Z)),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6989_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6990_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xa),Z))),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xa)),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6991_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( minus_minus_complex(uminus1482373934393186551omplex(divide1717551699836669952omplex(Xa,Z)),Y) = divide1717551699836669952omplex(minus_minus_complex(uminus1482373934393186551omplex(Xa),aa_complex_complex(times_times_complex(Y),Z)),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6992_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6993_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),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(A),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6994_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      minus_minus_complex(divide1717551699836669952omplex(A,Z),B) = $ite(Z = zero_zero_complex,uminus1482373934393186551omplex(B),divide1717551699836669952omplex(minus_minus_complex(A,aa_complex_complex(times_times_complex(B),Z)),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6995_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6996_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),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,A)),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6997_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      minus_minus_complex(uminus1482373934393186551omplex(divide1717551699836669952omplex(A,Z)),B) = $ite(Z = zero_zero_complex,uminus1482373934393186551omplex(B),divide1717551699836669952omplex(minus_minus_complex(uminus1482373934393186551omplex(A),aa_complex_complex(times_times_complex(B),Z)),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6998_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z))),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6999_power2__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xa = Y )
        | ( Xa = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_7000_power2__eq__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xa = Y )
        | ( Xa = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_7001_power2__eq__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xa = Y )
        | ( Xa = uminus1351360451143612070nteger(Y) ) ) ) ).

% power2_eq_iff
tff(fact_7002_power2__eq__iff,axiom,
    ! [Xa: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xa = Y )
        | ( Xa = uminus1482373934393186551omplex(Y) ) ) ) ).

% power2_eq_iff
tff(fact_7003_power2__eq__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xa = Y )
        | ( Xa = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% power2_eq_iff
tff(fact_7004_lemma__interval,axiom,
    ! [A: real,Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xa),Y4))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_7005_round__diff__minimal,axiom,
    ! [Z: real,M: 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,M)))) ).

% round_diff_minimal
tff(fact_7006_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z),aa_int_rat(ring_1_of_int_rat,M)))) ).

% round_diff_minimal
tff(fact_7007_abs__le__square__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7008_abs__le__square__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xa)),abs_abs_Code_integer(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7009_abs__le__square__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xa)),aa_rat_rat(abs_abs_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7010_abs__le__square__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xa)),aa_int_int(abs_abs_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7011_abs__square__eq__1,axiom,
    ! [Xa: int] :
      ( ( aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xa) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_7012_abs__square__eq__1,axiom,
    ! [Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,Xa) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_7013_abs__square__eq__1,axiom,
    ! [Xa: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(Xa) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_7014_abs__square__eq__1,axiom,
    ! [Xa: rat] :
      ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,Xa) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_7015_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> 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(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_7016_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_7017_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(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(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_7018_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_7019_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_7020_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_7021_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(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(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_7022_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_7023_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_7024_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_7025_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(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(A),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(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_7026_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_7027_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_7028_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(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,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(aa_rat_fun_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_7029_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_7030_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_rat_rat(uminus_uminus_rat,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(aa_rat_fun_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(aa_rat_fun_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_7031_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_7032_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_7033_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_7034_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = uminus1482373934393186551omplex(one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_7035_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_7036_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( 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,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7037_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7038_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7039_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7040_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( 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,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7041_realpow__square__minus__le,axiom,
    ! [U: real,Xa: 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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_7042_powr__neg__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(divide_divide_real(one_one_real),Xa) ) ) ).

% powr_neg_one
tff(fact_7043_ln__add__one__self__le__self2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa) ) ).

% ln_add_one_self_le_self2
tff(fact_7044_power2__le__iff__abs__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7045_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xa)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7046_power2__le__iff__abs__le,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xa)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7047_power2__le__iff__abs__le,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xa)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7048_abs__square__le__1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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,Xa)),one_one_real) ) ).

% abs_square_le_1
tff(fact_7049_abs__square__le__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),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(Xa)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_7050_abs__square__le__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),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,Xa)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_7051_abs__square__le__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),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,Xa)),one_one_int) ) ).

% abs_square_le_1
tff(fact_7052_abs__square__less__1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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,Xa)),one_one_real) ) ).

% abs_square_less_1
tff(fact_7053_abs__square__less__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xa),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,Xa)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_7054_abs__square__less__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xa),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,Xa)),one_one_int) ) ).

% abs_square_less_1
tff(fact_7055_abs__square__less__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),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(Xa)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_7056_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_7057_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(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,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(aa_rat_fun_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_7058_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_7059_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_rat_rat(uminus_uminus_rat,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(aa_rat_fun_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(aa_rat_fun_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_7060_square__le__1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_7061_square__le__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_7062_square__le__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_7063_square__le__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_7064_minus__power__mult__self,axiom,
    ! [A: real,Na: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% minus_power_mult_self
tff(fact_7065_minus__power__mult__self,axiom,
    ! [A: int,Na: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% minus_power_mult_self
tff(fact_7066_minus__power__mult__self,axiom,
    ! [A: code_integer,Na: nat] : times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Na),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% minus_power_mult_self
tff(fact_7067_minus__power__mult__self,axiom,
    ! [A: complex,Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(A)),Na)),aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(A)),Na)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% minus_power_mult_self
tff(fact_7068_minus__power__mult__self,axiom,
    ! [A: rat,Na: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Na)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Na)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% minus_power_mult_self
tff(fact_7069_Bernoulli__inequality,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),Na)) ) ).

% Bernoulli_inequality
tff(fact_7070_ln__one__minus__pos__upper__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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),Xa))),aa_real_real(uminus_uminus_real,Xa)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_7071_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_7072_log__minus__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(minus_minus_real(aa_real_real(log(B),Xa)),Y) = aa_real_real(log(B),aa_real_real(times_times_real(Xa),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_7073_powr__neg__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,Na))) ) ) ).

% powr_neg_numeral
tff(fact_7074_of__int__round__abs__le,axiom,
    ! [Xa: 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,Xa))),Xa))),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_7075_of__int__round__abs__le,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))),Xa))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7076_round__unique_H,axiom,
    ! [Xa: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xa),aa_int_real(ring_1_of_int_real,Na)))),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,Xa) = Na ) ) ).

% round_unique'
tff(fact_7077_round__unique_H,axiom,
    ! [Xa: rat,Na: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_int_rat(ring_1_of_int_rat,Na)))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,Xa) = Na ) ) ).

% round_unique'
tff(fact_7078_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_7079_arsinh__def,axiom,
    ! [Xa: real] : aa_real_real(arsinh_real,Xa) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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_7080_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(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_7081_of__int__code__if,axiom,
    ! [K: 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,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))),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_7082_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),
          uminus1351360451143612070nteger(aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: code_integer,
            l:= 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_7083_of__int__code__if,axiom,
    ! [K: int] :
      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),
          uminus1482373934393186551omplex(ring_17405671764205052669omplex(aa_int_int(uminus_uminus_int,K))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),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,plus_plus_complex(l,one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_7084_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(aa_rat_fun_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(aa_rat_fun_rat_rat(plus_plus_rat,l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_7085_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_7086_of__real__neg__numeral,axiom,
    ! [W: num] : real_V4546457046886955230omplex(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_7087_of__real__of__nat__eq,axiom,
    ! [Na: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% of_real_of_nat_eq
tff(fact_7088_of__real__of__nat__eq,axiom,
    ! [Na: nat] : real_V4546457046886955230omplex(aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_complex(semiri8010041392384452111omplex,Na) ).

% of_real_of_nat_eq
tff(fact_7089_of__real__power,axiom,
    ! [Xa: real,Na: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1803761363581548252l_real,Xa)),Na) ).

% of_real_power
tff(fact_7090_of__real__power,axiom,
    ! [Xa: real,Na: nat] : real_V4546457046886955230omplex(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_nat_complex(power_power_complex(real_V4546457046886955230omplex(Xa)),Na) ).

% of_real_power
tff(fact_7091_of__real__add,axiom,
    ! [Xa: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xa)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_7092_of__real__add,axiom,
    ! [Xa: real,Y: real] : real_V4546457046886955230omplex(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) = plus_plus_complex(real_V4546457046886955230omplex(Xa),real_V4546457046886955230omplex(Y)) ).

% of_real_add
tff(fact_7093_of__real__divide,axiom,
    ! [Xa: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(Xa),Y)) = aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,Xa)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_divide
tff(fact_7094_of__real__divide,axiom,
    ! [Xa: real,Y: real] : real_V4546457046886955230omplex(aa_real_real(divide_divide_real(Xa),Y)) = divide1717551699836669952omplex(real_V4546457046886955230omplex(Xa),real_V4546457046886955230omplex(Y)) ).

% of_real_divide
tff(fact_7095_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(A2)),uminus5710092332889474511et_nat(B2))
    <=> aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ).

% Compl_subset_Compl_iff
tff(fact_7096_Compl__anti__mono,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(B2)),uminus5710092332889474511et_nat(A2)) ) ).

% Compl_anti_mono
tff(fact_7097_Compl__disjoint2,axiom,
    ! [A2: set_real] : inf_inf_set_real(uminus612125837232591019t_real(A2),A2) = bot_bot_set_real ).

% Compl_disjoint2
tff(fact_7098_Compl__disjoint2,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A2)),A2) = bot_bot_set_nat ).

% Compl_disjoint2
tff(fact_7099_Compl__disjoint2,axiom,
    ! [A2: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(A2),A2) = bot_bot_set_int ).

% Compl_disjoint2
tff(fact_7100_Compl__disjoint,axiom,
    ! [A2: set_real] : inf_inf_set_real(A2,uminus612125837232591019t_real(A2)) = bot_bot_set_real ).

% Compl_disjoint
tff(fact_7101_Compl__disjoint,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),uminus5710092332889474511et_nat(A2)) = bot_bot_set_nat ).

% Compl_disjoint
tff(fact_7102_Compl__disjoint,axiom,
    ! [A2: set_int] : inf_inf_set_int(A2,uminus1532241313380277803et_int(A2)) = bot_bot_set_int ).

% Compl_disjoint
tff(fact_7103_Diff__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),uminus5710092332889474511et_nat(B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% Diff_Compl
tff(fact_7104_Compl__Diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A2)),B2) ).

% Compl_Diff_eq
tff(fact_7105_subset__Compl__singleton,axiom,
    ! [A2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),uminus8041839845116263051T_VEBT(insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT)))
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),A2) ) ).

% subset_Compl_singleton
tff(fact_7106_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(insert_real(B,bot_bot_set_real)))
    <=> ~ aa_set_real_o(member_real(B),A2) ) ).

% subset_Compl_singleton
tff(fact_7107_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B),A2) ) ).

% subset_Compl_singleton
tff(fact_7108_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_7109_negative__eq__positive,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_7110_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_7111_of__real__numeral,axiom,
    ! [W: num] : real_V4546457046886955230omplex(aa_num_real(numeral_numeral_real,W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_7112_of__real__mult,axiom,
    ! [Xa: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(times_times_real(Xa),Y)) = aa_real_real(times_times_real(aa_real_real(real_V1803761363581548252l_real,Xa)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_mult
tff(fact_7113_of__real__mult,axiom,
    ! [Xa: real,Y: real] : real_V4546457046886955230omplex(aa_real_real(times_times_real(Xa),Y)) = aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(Xa)),real_V4546457046886955230omplex(Y)) ).

% of_real_mult
tff(fact_7114_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_7115_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_7116_nat__zminus__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_7117_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_7118_int__div__minus__is__minus1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = aa_int_int(uminus_uminus_int,A) )
      <=> ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_7119_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),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,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_7120_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),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,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_7121_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_7122_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_7123_int__cases2,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,N)) ) ).

% int_cases2
tff(fact_7124_Collect__imp__eq,axiom,
    ! [P: fun_int_o,Q: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dl(P),Q)) = sup_sup_set_int(uminus1532241313380277803et_int(collect_int(P)),collect_int(Q)) ).

% Collect_imp_eq
tff(fact_7125_Collect__imp__eq,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] : collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dm(P),Q)) = sup_sup_set_complex(uminus8566677241136511917omplex(collect_complex(P)),collect_complex(Q)) ).

% Collect_imp_eq
tff(fact_7126_Collect__imp__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_dn(P),Q)) = sup_su6024340866399070445nt_int(uminus6221592323253981072nt_int(collec213857154873943460nt_int(P)),collec213857154873943460nt_int(Q)) ).

% Collect_imp_eq
tff(fact_7127_Collect__imp__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_do(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(collect_nat(P))),collect_nat(Q)) ).

% Collect_imp_eq
tff(fact_7128_abs__zmult__eq__1,axiom,
    ! [M: int,Na: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),Na)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7129_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(A2))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_7130_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_7131_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),uminus5710092332889474511et_nat(A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_7132_Compl__Un,axiom,
    ! [A2: set_nat,B2: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A2)),uminus5710092332889474511et_nat(B2)) ).

% Compl_Un
tff(fact_7133_Compl__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A2)),uminus5710092332889474511et_nat(B2)) ).

% Compl_Int
tff(fact_7134_zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Na) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( Na = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Na = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_7135_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,Na: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Na) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_7136_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_7137_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_7138_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_7139_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_7140_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_7141_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_7142_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_7143_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(A2,B2) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_7144_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A2),uminus5710092332889474511et_nat(B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_7145_Compl__insert,axiom,
    ! [Xa: vEBT_VEBT,A2: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(insert_VEBT_VEBT(Xa,A2)) = minus_5127226145743854075T_VEBT(uminus8041839845116263051T_VEBT(A2),insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT)) ).

% Compl_insert
tff(fact_7146_Compl__insert,axiom,
    ! [Xa: real,A2: set_real] : uminus612125837232591019t_real(insert_real(Xa,A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),insert_real(Xa,bot_bot_set_real)) ).

% Compl_insert
tff(fact_7147_Compl__insert,axiom,
    ! [Xa: int,A2: set_int] : uminus1532241313380277803et_int(insert_int(Xa,A2)) = minus_minus_set_int(uminus1532241313380277803et_int(A2),insert_int(Xa,bot_bot_set_int)) ).

% Compl_insert
tff(fact_7148_Compl__insert,axiom,
    ! [Xa: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat(Xa,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,uminus5710092332889474511et_nat(A2)),insert_nat(Xa,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_7149_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_7150_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7151_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_7152_negative__zle__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),zero_zero_int) ).

% negative_zle_0
tff(fact_7153_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% zmod_zminus1_eq_if
tff(fact_7154_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ).

% zmod_zminus2_eq_if
tff(fact_7155_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_7156_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_7157_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_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(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)))) ).

% mod_abs_eq_div_nat
tff(fact_7158_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_7159_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Na))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_7160_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_7161_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,A)),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,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_7162_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_int_rat(ring_1_of_int_rat,A)),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,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_7163_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% nat_abs_int_diff
tff(fact_7164_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_7165_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: 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),Z4)) = 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,Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_7166_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_7167_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_7168_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_7169_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_7170_incr__lemma,axiom,
    ! [D: int,Z: int,Xa: 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,Xa),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,Xa),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_7171_decr__lemma,axiom,
    ! [D: int,Xa: 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,Xa),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,Xa),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_7172_minus__1__div__exp__eq__int,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_7173_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_7174_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K2: int] :
              ( aa_int_o(P,K2)
             => ( ( K2 != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K2: int] :
                ( aa_int_o(P,K2)
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_7175_m1mod2k,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int) ).

% m1mod2k
tff(fact_7176_nat0__intermed__int__val,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7177_sb__dec__lem_H,axiom,
    ! [K: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A)) ) ).

% sb_dec_lem'
tff(fact_7178_m1mod22k,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),one_one_int) ).

% m1mod22k
tff(fact_7179_sb__dec__lem,axiom,
    ! [K: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A)) ) ).

% sb_dec_lem
tff(fact_7180_powr__int,axiom,
    ! [Xa: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),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,Xa),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,Xa),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_7181_nonzero__of__real__divide,axiom,
    ! [Y: real,Xa: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(Xa),Y)) = aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,Xa)),aa_real_real(real_V1803761363581548252l_real,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_7182_nonzero__of__real__divide,axiom,
    ! [Y: real,Xa: real] :
      ( ( Y != zero_zero_real )
     => ( real_V4546457046886955230omplex(aa_real_real(divide_divide_real(Xa),Y)) = divide1717551699836669952omplex(real_V4546457046886955230omplex(Xa),real_V4546457046886955230omplex(Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_7183_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xa: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => aa_real_o(aa_real_fun_real_o(P,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),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,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7184_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xa: code_integer] :
      ( ! [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7185_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xa: rat] :
      ( ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
         => aa_rat_o(aa_rat_fun_rat_o(P,X),aa_nat_rat(power_power_rat(X),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,Xa)),aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7186_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xa: int] :
      ( ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => aa_int_o(aa_int_fun_int_o(P,X),aa_nat_int(power_power_int(X),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,Xa)),aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7187_monoseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_dp,Xa)) ) ).

% monoseq_arctan_series
tff(fact_7188_htt__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_predi(Tia,Xa),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_dq(T),Tia),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(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% htt_vebt_predi
tff(fact_7189_htt__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_succi(Tia,Xa),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_dr(T),Tia),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(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% htt_vebt_succi
tff(fact_7190_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_7191_dbl__dec__simps_I4_J,axiom,
    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_7192_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7193_dbl__dec__simps_I4_J,axiom,
    neg_nu6511756317524482435omplex(uminus1482373934393186551omplex(one_one_complex)) = uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7194_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_7195_signed__take__bit__rec,axiom,
    ! [Na: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Na,A) = $ite(Na = zero_zero_nat,uminus1351360451143612070nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),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,Na),one_one_nat),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_7196_signed__take__bit__rec,axiom,
    ! [Na: nat,A: int] :
      bit_ri631733984087533419it_int(Na,A) = $ite(Na = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_7197_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A2))
    <=> ~ aa_set_nat_o(member_nat(C),A2) ) ).

% Compl_iff
tff(fact_7198_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( aa_set_real_o(member_real(C),uminus612125837232591019t_real(A2))
    <=> ~ aa_set_real_o(member_real(C),A2) ) ).

% Compl_iff
tff(fact_7199_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A2))
    <=> ~ aa_set_int_o(member_int(C),A2) ) ).

% Compl_iff
tff(fact_7200_Compl__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),uminus8041839845116263051T_VEBT(A2))
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2) ) ).

% Compl_iff
tff(fact_7201_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(C),A2)
     => aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A2)) ) ).

% ComplI
tff(fact_7202_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real(C),A2)
     => aa_set_real_o(member_real(C),uminus612125837232591019t_real(A2)) ) ).

% ComplI
tff(fact_7203_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(C),A2)
     => aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A2)) ) ).

% ComplI
tff(fact_7204_ComplI,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),uminus8041839845116263051T_VEBT(A2)) ) ).

% ComplI
tff(fact_7205_signed__take__bit__of__0,axiom,
    ! [Na: nat] : bit_ri631733984087533419it_int(Na,zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_7206_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_7207_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_7208_dbl__dec__simps_I3_J,axiom,
    neg_nu3811975205180677377ec_int(one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_7209_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_7210_signed__take__bit__of__minus__1,axiom,
    ! [Na: nat] : bit_ri6519982836138164636nteger(Na,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_7211_signed__take__bit__of__minus__1,axiom,
    ! [Na: nat] : bit_ri631733984087533419it_int(Na,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_7212_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_7213_dbl__dec__simps_I2_J,axiom,
    neg_nu3811975205180677377ec_int(zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_7214_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_7215_dbl__dec__simps_I2_J,axiom,
    neg_nu6511756317524482435omplex(zero_zero_complex) = uminus1482373934393186551omplex(one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_7216_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_7217_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7218_signed__take__bit__0,axiom,
    ! [A: int] : bit_ri631733984087533419it_int(zero_zero_nat,A) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7219_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus2572480474538279996real_o(aTP_Lamm_a(A2))) ).

% uminus_set_def
tff(fact_7220_uminus__set__def,axiom,
    ! [A2: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(A2) = collect_VEBT_VEBT(uminus3031588813561569520VEBT_o(aTP_Lamm_aa(A2))) ).

% uminus_set_def
tff(fact_7221_uminus__set__def,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ab(A2))) ).

% uminus_set_def
tff(fact_7222_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_ac(A2))) ).

% uminus_set_def
tff(fact_7223_uminus__set__def,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(uminus4035047602201070266plex_o(aTP_Lamm_ad(A2))) ).

% uminus_set_def
tff(fact_7224_uminus__set__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A2) = collec213857154873943460nt_int(uminus103585356917985579_int_o(aTP_Lamm_ae(A2))) ).

% uminus_set_def
tff(fact_7225_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A2))
     => ~ aa_set_nat_o(member_nat(C),A2) ) ).

% ComplD
tff(fact_7226_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( aa_set_real_o(member_real(C),uminus612125837232591019t_real(A2))
     => ~ aa_set_real_o(member_real(C),A2) ) ).

% ComplD
tff(fact_7227_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A2))
     => ~ aa_set_int_o(member_int(C),A2) ) ).

% ComplD
tff(fact_7228_ComplD,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),uminus8041839845116263051T_VEBT(A2))
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2) ) ).

% ComplD
tff(fact_7229_Collect__neg__eq,axiom,
    ! [P: fun_nat_o] : collect_nat(aTP_Lamm_ds(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_7230_Collect__neg__eq,axiom,
    ! [P: fun_int_o] : collect_int(aTP_Lamm_dt(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_7231_Collect__neg__eq,axiom,
    ! [P: fun_complex_o] : collect_complex(aTP_Lamm_du(P)) = uminus8566677241136511917omplex(collect_complex(P)) ).

% Collect_neg_eq
tff(fact_7232_Collect__neg__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aTP_Lamm_dv(P)) = uminus6221592323253981072nt_int(collec213857154873943460nt_int(P)) ).

% Collect_neg_eq
tff(fact_7233_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_dw(A2)) ).

% Compl_eq
tff(fact_7234_Compl__eq,axiom,
    ! [A2: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(A2) = collect_VEBT_VEBT(aTP_Lamm_dx(A2)) ).

% Compl_eq
tff(fact_7235_Compl__eq,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(aTP_Lamm_dy(A2)) ).

% Compl_eq
tff(fact_7236_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_dz(A2)) ).

% Compl_eq
tff(fact_7237_Compl__eq,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(aTP_Lamm_ea(A2)) ).

% Compl_eq
tff(fact_7238_Compl__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A2) = collec213857154873943460nt_int(aTP_Lamm_eb(A2)) ).

% Compl_eq
tff(fact_7239_signed__take__bit__minus,axiom,
    ! [Na: nat,K: int] : bit_ri631733984087533419it_int(Na,aa_int_int(uminus_uminus_int,bit_ri631733984087533419it_int(Na,K))) = bit_ri631733984087533419it_int(Na,aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_7240_signed__take__bit__add,axiom,
    ! [Na: nat,K: int,L: int] : bit_ri631733984087533419it_int(Na,aa_int_int(aa_int_fun_int_int(plus_plus_int,bit_ri631733984087533419it_int(Na,K)),bit_ri631733984087533419it_int(Na,L))) = bit_ri631733984087533419it_int(Na,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_7241_signed__take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : bit_ri631733984087533419it_int(Na,aa_int_int(aa_int_fun_int_int(times_times_int,bit_ri631733984087533419it_int(Na,K)),bit_ri631733984087533419it_int(Na,L))) = bit_ri631733984087533419it_int(Na,aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% signed_take_bit_mult
tff(fact_7242_signed__take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : bit_ri631733984087533419it_int(Na,aa_int_int(aa_int_fun_int_int(minus_minus_int,bit_ri631733984087533419it_int(Na,K)),bit_ri631733984087533419it_int(Na,L))) = bit_ri631733984087533419it_int(Na,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_7243_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,bit_ri631733984087533419it_int(Na,K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_7244_monoseq__realpow,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xa)) ) ) ).

% monoseq_realpow
tff(fact_7245_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_ri631733984087533419it_int(Na,K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_7246_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),bit_ri631733984087533419it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_7247_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_ri631733984087533419it_int(Na,K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_7248_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),bit_ri631733984087533419it_int(Na,K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_7249_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),bit_ri631733984087533419it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_7250_dbl__dec__def,axiom,
    ! [Xa: real] : aa_real_real(neg_nu6075765906172075777c_real,Xa) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Xa)),one_one_real) ).

% dbl_dec_def
tff(fact_7251_dbl__dec__def,axiom,
    ! [Xa: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,Xa) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Xa)),one_one_rat) ).

% dbl_dec_def
tff(fact_7252_dbl__dec__def,axiom,
    ! [Xa: int] : neg_nu3811975205180677377ec_int(Xa) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xa)),one_one_int) ).

% dbl_dec_def
tff(fact_7253_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( bit_ri631733984087533419it_int(Na,K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_7254_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_ri631733984087533419it_int(Na,K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_7255_helpypredd,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% helpypredd
tff(fact_7256_helpyd,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_succ(T,Xa) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% helpyd
tff(fact_7257_pred__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ec(T),Xa)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_7258_succ__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_succ(T,Xa) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ed(T),Xa)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_7259_vebt__pred_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_vebt_predi(T,Tia,Xa),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_dq(T),Tia),Xa)) ) ).

% vebt_pred'_rf_abstr
tff(fact_7260_vebt__succi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_vebt_succi(T,Tia,Xa),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_dr(T),Tia),Xa)) ) ).

% vebt_succi'_rf_abstr
tff(fact_7261_power__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xa)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_7262_pred__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(T)),Xa),Sx) ) ) ).

% pred_correct
tff(fact_7263_set__vebt__pred,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Px: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some_nat(Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(T)),Xa),Px) ) ) ).

% set_vebt_pred
tff(fact_7264_succ__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_succ(T,Xa) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(T)),Xa),Sx) ) ) ).

% succ_correct
tff(fact_7265_pred__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Px: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some_nat(Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),Xa),Px) ) ) ).

% pred_corr
tff(fact_7266_succ__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_succ(T,Xa) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),Xa),Sx) ) ) ).

% succ_corr
tff(fact_7267_lesseq__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> vEBT_VEBT_lesseq(some_nat(Xa),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_7268_sup__Some,axiom,
    ! [Xa: set_nat,Y: set_nat] : sup_su3598758113090618626et_nat(some_set_nat(Xa),some_set_nat(Y)) = some_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Y)) ).

% sup_Some
tff(fact_7269_sup__Some,axiom,
    ! [Xa: nat,Y: nat] : sup_sup_option_nat(some_nat(Xa),some_nat(Y)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Y)) ).

% sup_Some
tff(fact_7270_sup__Some,axiom,
    ! [Xa: int,Y: int] : sup_sup_option_int(some_int(Xa),some_int(Y)) = some_int(aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Y)) ).

% sup_Some
tff(fact_7271_sup__Some,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : sup_su2190601396275437440d_enat(some_Extended_enat(Xa),some_Extended_enat(Y)) = some_Extended_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y)) ).

% sup_Some
tff(fact_7272_sup__Some,axiom,
    ! [Xa: assn,Y: assn] : sup_sup_option_assn(some_assn(Xa),some_assn(Y)) = some_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),Y)) ).

% sup_Some
tff(fact_7273_less__option__None__Some__code,axiom,
    ! [Xa: nat] : ord_less_option_nat(none_nat,some_nat(Xa)) ).

% less_option_None_Some_code
tff(fact_7274_less__option__None__Some__code,axiom,
    ! [Xa: num] : ord_less_option_num(none_num,some_num(Xa)) ).

% less_option_None_Some_code
tff(fact_7275_less__option__Some,axiom,
    ! [Xa: real,Y: real] :
      ( ord_less_option_real(some_real(Xa),some_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% less_option_Some
tff(fact_7276_less__option__Some,axiom,
    ! [Xa: rat,Y: rat] :
      ( ord_less_option_rat(some_rat(Xa),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ).

% less_option_Some
tff(fact_7277_less__option__Some,axiom,
    ! [Xa: num,Y: num] :
      ( ord_less_option_num(some_num(Xa),some_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y) ) ).

% less_option_Some
tff(fact_7278_less__option__Some,axiom,
    ! [Xa: nat,Y: nat] :
      ( ord_less_option_nat(some_nat(Xa),some_nat(Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ).

% less_option_Some
tff(fact_7279_less__option__Some,axiom,
    ! [Xa: int,Y: int] :
      ( ord_less_option_int(some_int(Xa),some_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ).

% less_option_Some
tff(fact_7280_less__option__Some,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ord_le7113747843092208513nteger(some_Code_integer(Xa),some_Code_integer(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y) ) ).

% less_option_Some
tff(fact_7281_misiz,axiom,
    ! [T: vEBT_VEBT,Na: nat,M: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( some_nat(M) = vEBT_vebt_mint(T) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% misiz
tff(fact_7282_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : 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,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_7283_vebt__minNull__mint,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
    <=> ( vEBT_vebt_mint(T) = none_nat ) ) ).

% vebt_minNull_mint
tff(fact_7284_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T) ) ).

% minminNull
tff(fact_7285_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_7286_mint__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(T),Maxi) ) ) ).

% mint_member
tff(fact_7287_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,Na: nat,Mini: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( aa_nat_o(vEBT_vebt_member(T),Xa)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xa) ) ) ) ).

% mint_corr_help
tff(fact_7288_set__vebt__mint_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Xa) )
      <=> ( aa_set_nat_o(member_nat(Xa),vEBT_set_vebt(T))
          & ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),vEBT_set_vebt(T))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4) ) ) ) ) ).

% set_vebt_mint'
tff(fact_7289_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_7290_set__vebt__mint,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Xa) )
      <=> aa_nat_o(vEBT_VEBT_min_in_set(vEBT_set_vebt(T)),Xa) ) ) ).

% set_vebt_mint
tff(fact_7291_mint__sound,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),Xa)
       => ( vEBT_vebt_mint(T) = some_nat(Xa) ) ) ) ).

% mint_sound
tff(fact_7292_mint__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Xa) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),Xa) ) ) ).

% mint_corr
tff(fact_7293_less__option__None,axiom,
    ! [Xa: option_nat] : ~ ord_less_option_nat(Xa,none_nat) ).

% less_option_None
tff(fact_7294_less__option__None,axiom,
    ! [Xa: option_num] : ~ ord_less_option_num(Xa,none_num) ).

% less_option_None
tff(fact_7295_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_7296_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_7297_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : 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,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_7298_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : 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,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_7299_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : 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,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_7300_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_7301_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_7302_bot__option__def,axiom,
    bot_bot_option_nat = none_nat ).

% bot_option_def
tff(fact_7303_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
tff(fact_7304_less__option__None__Some,axiom,
    ! [Xa: nat] : ord_less_option_nat(none_nat,some_nat(Xa)) ).

% less_option_None_Some
tff(fact_7305_less__option__None__Some,axiom,
    ! [Xa: num] : ord_less_option_num(none_num,some_num(Xa)) ).

% less_option_None_Some
tff(fact_7306_less__option__None__is__Some,axiom,
    ! [Xa: option_nat] :
      ( ord_less_option_nat(none_nat,Xa)
     => ? [Z3: nat] : Xa = some_nat(Z3) ) ).

% less_option_None_is_Some
tff(fact_7307_less__option__None__is__Some,axiom,
    ! [Xa: option_num] :
      ( ord_less_option_num(none_num,Xa)
     => ? [Z3: num] : Xa = some_num(Z3) ) ).

% less_option_None_is_Some
tff(fact_7308_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_minti(Tia),aa_VEB7611521244314475576t_assn(aTP_Lamm_ee(T),Tia)),one_one_nat) ).

% vebt_minti_hT
tff(fact_7309_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_minti(Tia),aa_VEB7611521244314475576t_assn(aTP_Lamm_ee(T),Tia)) ).

% vebt_minti_h
tff(fact_7310_set__vebt__mint_H_H,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( vEBT_vebt_mint(T) = $ite(vEBT_set_vebt(T) = bot_bot_set_nat,none_nat,some_nat(lattic8721135487736765967in_nat(vEBT_set_vebt(T)))) ) ) ).

% set_vebt_mint''
tff(fact_7311_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_7312_dbl__simps_I4_J,axiom,
    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_7313_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7314_dbl__simps_I4_J,axiom,
    neg_nu7009210354673126013omplex(uminus1482373934393186551omplex(one_one_complex)) = uminus1482373934393186551omplex(numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7315_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_7316_summable__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ef,Xa)) ) ).

% summable_arctan_series
tff(fact_7317_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),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,bit_ri631733984087533419it_int(Na,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_7318_even__odd__cases,axiom,
    ! [Xa: nat] :
      ( ! [N: nat] : Xa != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : Xa != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_7319_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_vebt_minti(T)),one_one_nat) ).

% TBOUND_vebt_minti
tff(fact_7320_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_7321_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_7322_lessI,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Na)) ).

% lessI
tff(fact_7323_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) ) ).

% Suc_mono
tff(fact_7324_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_eq
tff(fact_7325_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% add_Suc_right
tff(fact_7326_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% Suc_le_mono
tff(fact_7327_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ).

% diff_Suc_Suc
tff(fact_7328_Suc__diff__diff,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),K) ).

% Suc_diff_diff
tff(fact_7329_dbl__simps_I2_J,axiom,
    neg_nu7009210354673126013omplex(zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_7330_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_7331_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_7332_dbl__simps_I2_J,axiom,
    neg_numeral_dbl_int(zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_7333_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Na)) = zero_zero_nat ).

% power_0_Suc
tff(fact_7334_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Na)) = zero_zero_int ).

% power_0_Suc
tff(fact_7335_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Na)) = zero_zero_real ).

% power_0_Suc
tff(fact_7336_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Na)) = zero_zero_complex ).

% power_0_Suc
tff(fact_7337_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_nat_nat(suc,Na)) = zero_z3403309356797280102nteger ).

% power_0_Suc
tff(fact_7338_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Na)) = zero_zero_rat ).

% power_0_Suc
tff(fact_7339_power__Suc0__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_7340_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_7341_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_7342_power__Suc0__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_7343_power__Suc0__right,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_7344_power__Suc0__right,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_7345_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_7346_zero__less__Suc,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Na)) ).

% zero_less_Suc
tff(fact_7347_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_7348_one__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_7349_mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_7350_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( Xa = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_7351_power__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_7352_diff__Suc__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),one_one_nat) = Na ).

% diff_Suc_1
tff(fact_7353_mult__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ).

% mult_Suc_right
tff(fact_7354_signed__take__bit__Suc__1,axiom,
    ! [Na: nat] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_7355_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_7356_eq__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Na) )
    <=> ( aa_num_nat(pred_numeral,K) = Na ) ) ).

% eq_numeral_Suc
tff(fact_7357_Suc__eq__numeral,axiom,
    ! [Na: nat,K: num] :
      ( ( aa_nat_nat(suc,Na) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Na = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_7358_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_nu7009210354673126013omplex(numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_7359_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_7360_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_7361_dbl__simps_I5_J,axiom,
    ! [K: num] : 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_7362_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_7363_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_7364_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_numeral_dbl_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,neg_numeral_dbl_int(aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_7365_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) = uminus1351360451143612070nteger(neg_nu8804712462038260780nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_simps(1)
tff(fact_7366_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu7009210354673126013omplex(uminus1482373934393186551omplex(numera6690914467698888265omplex(K))) = uminus1482373934393186551omplex(neg_nu7009210354673126013omplex(numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_7367_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_7368_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_7369_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_7370_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_7371_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_7372_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ).

% of_nat_Suc
tff(fact_7373_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = plus_plus_complex(one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_7374_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% Suc_pred
tff(fact_7375_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Na) ) ) ).

% one_le_mult_iff
tff(fact_7376_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_7377_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_7378_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_7379_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self1
tff(fact_7380_Suc__mod__mult__self2,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self2
tff(fact_7381_Suc__mod__mult__self3,axiom,
    ! [K: nat,Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self3
tff(fact_7382_Suc__mod__mult__self4,axiom,
    ! [Na: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)),M))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self4
tff(fact_7383_Suc__numeral,axiom,
    ! [Na: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Na),one)) ).

% Suc_numeral
tff(fact_7384_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_7385_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_7386_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% less_numeral_Suc
tff(fact_7387_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_7388_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% le_numeral_Suc
tff(fact_7389_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_7390_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Na) ).

% diff_numeral_Suc
tff(fact_7391_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc
tff(fact_7392_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc'
tff(fact_7393_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_7394_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_7395_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_7396_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Na ) ) ).

% Suc_diff_1
tff(fact_7397_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Na: 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)),Na))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_7398_dbl__simps_I3_J,axiom,
    neg_nu7009210354673126013omplex(one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_7399_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_7400_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_7401_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_int(one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_7402_dbl__simps_I3_J,axiom,
    neg_nu8804712462038260780nteger(one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_7403_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_7404_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_7405_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_7406_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_7407_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_7408_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,bit_ri631733984087533419it_int(Na,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_7409_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_7410_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),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,bit_ri631733984087533419it_int(Na,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_7411_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),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,bit_ri631733984087533419it_int(Na,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_7412_Suc__inject,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xa) = aa_nat_nat(suc,Y) )
     => ( Xa = Y ) ) ).

% Suc_inject
tff(fact_7413_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa_nat_nat(suc,Na) ).

% n_not_Suc_n
tff(fact_7414_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_13: nat] : aa_nat_o(P,X_13)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_7415_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_7416_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_7417_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_7418_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_7419_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_7420_nat__induct,axiom,
    ! [P: fun_nat_o,Na: 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,Na) ) ) ).

% nat_induct
tff(fact_7421_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Na: nat] :
      ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X),zero_zero_nat)
     => ( ! [Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y2))
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X)),aa_nat_nat(suc,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,M),Na) ) ) ) ).

% diff_induct
tff(fact_7422_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_7423_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_7424_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_7425_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_7426_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ? [M4: nat] : Na = aa_nat_nat(suc,M4) ) ).

% not0_implies_Suc
tff(fact_7427_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) )
       => ~ ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J3)
             => ( K != aa_nat_nat(suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_7428_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_lessD
tff(fact_7429_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J3)
           => ( K != aa_nat_nat(suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_7430_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( ( aa_nat_nat(suc,M) != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_7431_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_7432_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% less_SucI
tff(fact_7433_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Na)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
            & aa_nat_o(P,I2) ) ) ) ).

% Ex_less_Suc
tff(fact_7434_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_7435_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_7436_Nat_OAll__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Na)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
           => aa_nat_o(P,I2) ) ) ) ).

% Nat.All_less_Suc
tff(fact_7437_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M)
    <=> ? [M5: nat] :
          ( ( M = aa_nat_nat(suc,M5) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M5) ) ) ).

% Suc_less_eq2
tff(fact_7438_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_7439_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_SucD
tff(fact_7440_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_7441_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(suc,I3))
       => ( ! [I3: nat,J3: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I3),J3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J3),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I3),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_7442_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => aa_nat_o(P,I3) )
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I3))
               => aa_nat_o(P,I3) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_7443_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_7444_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_7445_add__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% add_Suc
tff(fact_7446_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) ).

% add_Suc_shift
tff(fact_7447_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R2: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(R2,X),X)
       => ( ! [X: nat,Y2: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R2,X),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(R2,Y2),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R2,X),Z3) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R2,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R2,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_7448_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(P,M)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_7449_full__nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% full_nat_induct
tff(fact_7450_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_7451_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_7452_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        | ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_7453_Suc__le__D,axiom,
    ! [Na: nat,M6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M6)
     => ? [M4: nat] : M6 = aa_nat_nat(suc,M4) ) ).

% Suc_le_D
tff(fact_7454_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_SucI
tff(fact_7455_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_SucE
tff(fact_7456_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_leD
tff(fact_7457_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_7458_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na) )
    <=> ( M = Na ) ) ).

% Suc_mult_cancel1
tff(fact_7459_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),Na) ).

% mod_Suc_Suc_eq
tff(fact_7460_mod__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% mod_Suc_eq
tff(fact_7461_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Na: nat,N6: 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,Na),N6)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_7462_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Na: nat,N6: 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,Na),N6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_7463_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Na: nat,N6: 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,Na),N6)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Na)),aa_nat_num(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_7464_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Na: nat,N6: 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,Na),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_7465_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Na: nat,N6: 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,Na),N6)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_7466_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_Code_integer,Na: nat,N6: nat] :
      ( ! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N)),aa_nat_Code_integer(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,Na)),aa_nat_Code_integer(F,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_7467_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Na: nat,M: 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,Na)),aa_nat_real(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_7468_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Na: nat,M: 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,Na)),aa_nat_rat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_7469_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Na: nat,M: 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,Na)),aa_nat_num(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_7470_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Na: nat,M: 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,Na)),aa_nat_nat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_7471_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Na: nat,M: 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,Na)),aa_nat_int(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_7472_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_Code_integer,Na: nat,M: nat] :
      ( ! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N)),aa_nat_Code_integer(F,aa_nat_nat(suc,N)))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,Na)),aa_nat_Code_integer(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_7473_power__Suc,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Na)) ).

% power_Suc
tff(fact_7474_power__Suc,axiom,
    ! [A: code_integer,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Na)) = times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ).

% power_Suc
tff(fact_7475_power__Suc,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ).

% power_Suc
tff(fact_7476_power__Suc,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na)) ).

% power_Suc
tff(fact_7477_power__Suc,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ).

% power_Suc
tff(fact_7478_power__Suc,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na)) ).

% power_Suc
tff(fact_7479_power__Suc2,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Na)),A) ).

% power_Suc2
tff(fact_7480_power__Suc2,axiom,
    ! [A: code_integer,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Na)) = times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na),A) ).

% power_Suc2
tff(fact_7481_power__Suc2,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),A) ).

% power_Suc2
tff(fact_7482_power__Suc2,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Na)),A) ).

% power_Suc2
tff(fact_7483_power__Suc2,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),A) ).

% power_Suc2
tff(fact_7484_power__Suc2,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Na)),A) ).

% power_Suc2
tff(fact_7485_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,Na: nat,N6: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,Na)),aa_nat_set_nat(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_7486_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N6: 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,Na),N6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_7487_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N6: 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,Na),N6)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Na)),aa_nat_num(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_7488_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N6: 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,Na),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_7489_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N6: 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,Na),N6)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Na)),aa_nat_int(F,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_7490_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,Na: nat,N6: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N))),aa_nat_set_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N6)),aa_nat_set_nat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_7491_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N6: 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,Na),N6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N6)),aa_nat_rat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_7492_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N6: 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,Na),N6)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N6)),aa_nat_num(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_7493_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N6: 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,Na),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N6)),aa_nat_nat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_7494_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N6: 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,Na),N6)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N6)),aa_nat_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_7495_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)) != zero_zero_real ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_7496_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) != zero_zero_int ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_7497_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Na)) != zero_zero_nat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_7498_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Na)) != zero_zero_rat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_7499_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Na)) != zero_z3403309356797280102nteger ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_7500_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Na)) != zero_zero_complex ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_7501_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
            & aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% Ex_less_Suc2
tff(fact_7502_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ? [M3: nat] : Na = aa_nat_nat(suc,M3) ) ).

% gr0_conv_Suc
tff(fact_7503_All__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% All_less_Suc2
tff(fact_7504_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ? [M4: nat] : Na = aa_nat_nat(suc,M4) ) ).

% gr0_implies_Suc
tff(fact_7505_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( ( M = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M = aa_nat_nat(suc,J2) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_7506_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_7507_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_7508_dbl__def,axiom,
    ! [Xa: real] : aa_real_real(neg_numeral_dbl_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Xa) ).

% dbl_def
tff(fact_7509_dbl__def,axiom,
    ! [Xa: rat] : aa_rat_rat(neg_numeral_dbl_rat,Xa) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Xa) ).

% dbl_def
tff(fact_7510_dbl__def,axiom,
    ! [Xa: int] : neg_numeral_dbl_int(Xa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xa) ).

% dbl_def
tff(fact_7511_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_7512_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ~ ! [Q3: nat] : Na != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_7513_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M))) ).

% less_add_Suc1
tff(fact_7514_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I))) ).

% less_add_Suc2
tff(fact_7515_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ? [K3: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_7516_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ? [K2: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_7517_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na) ) ).

% Suc_leI
tff(fact_7518_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_eq
tff(fact_7519_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_7520_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_7521_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_lessD
tff(fact_7522_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_7523_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_Suc_eq_le
tff(fact_7524_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_7525_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_7526_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = A ) ) ).

% nat_in_between_eq(2)
tff(fact_7527_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = aa_nat_nat(suc,A) ) ) ).

% nat_in_between_eq(1)
tff(fact_7528_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_7529_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_7530_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat) ).

% Suc_eq_plus1
tff(fact_7531_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_7532_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Na) ).

% Suc_eq_plus1_left
tff(fact_7533_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_7534_Suc__div__le__mono,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),Na)) ).

% Suc_div_le_mono
tff(fact_7535_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% Suc_diff_le
tff(fact_7536_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_7537_mult__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ).

% mult_Suc
tff(fact_7538_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_7539_mod__Suc,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = Na,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))) ).

% mod_Suc
tff(fact_7540_mod__induct,axiom,
    ! [P: fun_nat_o,Na: nat,P3: nat,M: nat] :
      ( aa_nat_o(P,Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),P3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P3)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P3)
               => ( 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)),P3)) ) )
           => aa_nat_o(P,M) ) ) ) ) ).

% mod_induct
tff(fact_7541_mod__Suc__le__divisor,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_7542_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_7543_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_7544_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_7545_power__le__imp__le__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Na)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_7546_power__le__imp__le__base,axiom,
    ! [A: code_integer,Na: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Na))),aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,Na)))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_7547_power__le__imp__le__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Na)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_7548_power__le__imp__le__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Na)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_7549_power__le__imp__le__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Na)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_7550_power__inject__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Na)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_7551_power__inject__base,axiom,
    ! [A: code_integer,Na: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,Na)) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_7552_power__inject__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Na)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_7553_power__inject__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Na)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_7554_power__inject__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Na)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_7555_power__gt1,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_7556_power__gt1,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_7557_power__gt1,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_7558_power__gt1,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_7559_power__gt1,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_7560_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_7561_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))) ).

% eval_nat_numeral(3)
tff(fact_7562_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K2)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_7563_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( 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,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_7564_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_7565_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% one_less_mult
tff(fact_7566_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_7567_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)) ) ) ).

% n_less_n_mult_m
tff(fact_7568_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),K)) ) ).

% power_gt_expt
tff(fact_7569_nat__one__le__power,axiom,
    ! [I: nat,Na: 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),Na)) ) ).

% nat_one_le_power
tff(fact_7570_realpow__pos__nth2,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R4),aa_nat_nat(suc,Na)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_7571_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_7572_int__Suc,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) ).

% int_Suc
tff(fact_7573_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N2: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% zless_iff_Suc_zadd
tff(fact_7574_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% atLeast0_atMost_Suc
tff(fact_7575_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_7576_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_7577_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Na)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_7578_power__Suc__le__self,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_7579_power__Suc__le__self,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_7580_power__Suc__le__self,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_7581_power__Suc__le__self,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_7582_power__Suc__le__self,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_7583_power__Suc__less__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_7584_power__Suc__less__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_7585_power__Suc__less__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_7586_power__Suc__less__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_7587_power__Suc__less__one,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Na))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_7588_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_7589_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_7590_Suc__double__not__eq__double,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% Suc_double_not_eq_double
tff(fact_7591_double__not__eq__Suc__double,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% double_not_eq_Suc_double
tff(fact_7592_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na) ).

% Suc3_eq_add_3
tff(fact_7593_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M) ) ) ).

% nz_le_conv_less
tff(fact_7594_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_7595_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_7596_div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_7597_div__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ) ).

% div_if
tff(fact_7598_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = $ite(M = zero_zero_nat,Na,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Na))) ).

% add_eq_if
tff(fact_7599_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_7600_div__nat__eqI,axiom,
    ! [Na: nat,Q2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q2)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_7601_Suc__nat__number__of__add,axiom,
    ! [V: num,Na: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),Na) ).

% Suc_nat_number_of_add
tff(fact_7602_not__zle__0__negative,axiom,
    ! [Na: 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,Na)))) ).

% not_zle_0_negative
tff(fact_7603_negD,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
     => ? [N: nat] : Xa = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_7604_negative__zless__0,axiom,
    ! [Na: 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,Na)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7605_Suc__as__int,axiom,
    ! [X5: nat] : aa_nat_nat(suc,X5) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),one_one_int)) ).

% Suc_as_int
tff(fact_7606_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_7607_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(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_7608_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_7609_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_7610_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Na) ).

% Suc_div_eq_add3_div
tff(fact_7611_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_7612_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_7613_split__div_H,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q5)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q5)))
            & aa_nat_o(P,Q5) ) ) ) ).

% split_div'
tff(fact_7614_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_7615_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_7616_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_7617_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_7618_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_7619_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_7620_VEBT__internal_OTb_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,zero_zero_nat)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_7621_power__odd__eq,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_7622_power__odd__eq,axiom,
    ! [A: code_integer,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_7623_power__odd__eq,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_7624_power__odd__eq,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_7625_power__odd__eq,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_7626_power__odd__eq,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_7627_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => 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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_7628_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => 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,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_7629_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Na: 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,Na) ) ) ) ).

% nat_bit_induct
tff(fact_7630_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
           => ? [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I3)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7631_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_7632_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_7633_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_7634_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_7635_odd__power__less__zero,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_7636_odd__power__less__zero,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_7637_odd__power__less__zero,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_7638_odd__power__less__zero,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_z3403309356797280102nteger) ) ).

% odd_power_less_zero
tff(fact_7639_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_7640_power__minus1__odd,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_7641_power__minus1__odd,axiom,
    ! [Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(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))),Na))) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% power_minus1_odd
tff(fact_7642_power__minus1__odd,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_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))),Na))) = uminus1482373934393186551omplex(one_one_complex) ).

% power_minus1_odd
tff(fact_7643_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_7644_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_7645_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_ri631733984087533419it_int(Na,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_7646_nat__ivt__aux,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7647_signed__take__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),bit_ri6519982836138164636nteger(Na,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_7648_signed__take__bit__Suc,axiom,
    ! [Na: nat,A: int] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_ri631733984087533419it_int(Na,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_7649_nat__div__eq__Suc__0__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_7650_unset__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),bit_se8260200283734997820nteger(Na,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_7651_unset__bit__Suc,axiom,
    ! [Na: nat,A: int] : bit_se4203085406695923979it_int(aa_nat_nat(suc,Na),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se4203085406695923979it_int(Na,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_7652_unset__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Na)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_7653_set__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),bit_se2793503036327961859nteger(Na,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_7654_set__bit__Suc,axiom,
    ! [Na: nat,A: int] : bit_se7879613467334960850it_int(aa_nat_nat(suc,Na),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se7879613467334960850it_int(Na,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_7655_set__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Na)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_7656_sb__inc__lem,axiom,
    ! [A: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem
tff(fact_7657_sb__inc__lem_H,axiom,
    ! [A: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem'
tff(fact_7658_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
     => 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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na)))),bit_ri631733984087533419it_int(Na,K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_7659_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_eg(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7660_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_eh(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7661_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ei(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7662_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ej(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7663_Min__singleton,axiom,
    ! [Xa: real] : lattic3629708407755379051n_real(insert_real(Xa,bot_bot_set_real)) = Xa ).

% Min_singleton
tff(fact_7664_Min__singleton,axiom,
    ! [Xa: int] : lattic8718645017227715691in_int(insert_int(Xa,bot_bot_set_int)) = Xa ).

% Min_singleton
tff(fact_7665_Min__singleton,axiom,
    ! [Xa: nat] : lattic8721135487736765967in_nat(insert_nat(Xa,bot_bot_set_nat)) = Xa ).

% Min_singleton
tff(fact_7666_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_7667_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_el(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7668_summable__iff__shift,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_em(F),K))
    <=> summable_complex(F) ) ).

% summable_iff_shift
tff(fact_7669_summable__zero,axiom,
    summable_complex(aTP_Lamm_en) ).

% summable_zero
tff(fact_7670_summable__zero,axiom,
    summable_real(aTP_Lamm_eo) ).

% summable_zero
tff(fact_7671_summable__zero,axiom,
    summable_nat(aTP_Lamm_cb) ).

% summable_zero
tff(fact_7672_summable__zero,axiom,
    summable_int(aTP_Lamm_ep) ).

% summable_zero
tff(fact_7673_summable__complex__of__real,axiom,
    ! [F: fun_nat_real] :
      ( summable_complex(aTP_Lamm_eq(F))
    <=> summable_real(F) ) ).

% summable_complex_of_real
tff(fact_7674_summable__single,axiom,
    ! [I: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_er(I),F)) ).

% summable_single
tff(fact_7675_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_es(I),F)) ).

% summable_single
tff(fact_7676_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_et(I),F)) ).

% summable_single
tff(fact_7677_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_eu(I),F)) ).

% summable_single
tff(fact_7678_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aTP_Lamm_ev(C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_7679_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ew,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7680_summable__add,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ex(F),G)) ) ) ).

% summable_add
tff(fact_7681_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ey(F),G)) ) ) ).

% summable_add
tff(fact_7682_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ez(F),G)) ) ) ).

% summable_add
tff(fact_7683_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_fa(F),G)) ) ) ).

% summable_add
tff(fact_7684_summable__mult2,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fb(F),C)) ) ).

% summable_mult2
tff(fact_7685_summable__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_fc(F),C)) ) ).

% summable_mult2
tff(fact_7686_summable__mult,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fd(F),C)) ) ).

% summable_mult
tff(fact_7687_summable__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_fe(F),C)) ) ).

% summable_mult
tff(fact_7688_summable__diff,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ff(F),G)) ) ) ).

% summable_diff
tff(fact_7689_summable__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fg(F),G)) ) ) ).

% summable_diff
tff(fact_7690_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fh(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_7691_summable__Suc__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_fi(F))
    <=> summable_complex(F) ) ).

% summable_Suc_iff
tff(fact_7692_summable__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_eg(F),C)) ) ).

% summable_divide
tff(fact_7693_summable__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_eh(F),C)) ) ).

% summable_divide
tff(fact_7694_summable__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => summable_real(aTP_Lamm_fj(F)) ) ).

% summable_minus
tff(fact_7695_summable__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_complex(aTP_Lamm_fk(F)) ) ).

% summable_minus
tff(fact_7696_summable__minus__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fj(F))
    <=> summable_real(F) ) ).

% summable_minus_iff
tff(fact_7697_summable__minus__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_fk(F))
    <=> summable_complex(F) ) ).

% summable_minus_iff
tff(fact_7698_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_el(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7699_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(F)
     => summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_em(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7700_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fl(F))
     => summable_real(F) ) ).

% summable_rabs_cancel
tff(fact_7701_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ei(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_7702_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ej(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7703_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7704_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_7705_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_7706_summable__of__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => summable_real(aTP_Lamm_fm(X7)) ) ).

% summable_of_real
tff(fact_7707_summable__of__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => summable_complex(aTP_Lamm_eq(X7)) ) ).

% summable_of_real
tff(fact_7708_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_fn(F)) ).

% summable_zero_power'
tff(fact_7709_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_fo(F)) ).

% summable_zero_power'
tff(fact_7710_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_fp(F)) ).

% summable_zero_power'
tff(fact_7711_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_fn(F)) ).

% summable_0_powser
tff(fact_7712_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_fo(F)) ).

% summable_0_powser
tff(fact_7713_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fq(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7714_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fs(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7715_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fq(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7716_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_fs(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7717_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,M: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ft(F),M),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7718_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,M: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fu(F),M),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7719_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),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_fl(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7720_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K6: real,Na: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(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(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7721_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K6: real,Na: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(Z)),K6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(plus_plus_complex(Z,H))),K6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(minus_minus_complex(divide1717551699836669952omplex(minus_minus_complex(aa_nat_complex(power_power_complex(plus_plus_complex(Z,H)),Na),aa_nat_complex(power_power_complex(Z),Na)),H),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),real_V1022390504157884413omplex(H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7722_ln__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xa) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_fv,Xa)) ) ) ) ).

% ln_series
tff(fact_7723_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != one_one_int ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_7724_flip__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),bit_se1345352211410354436nteger(Na,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_7725_flip__bit__Suc,axiom,
    ! [Na: nat,A: int] : bit_se2159334234014336723it_int(aa_nat_nat(suc,Na),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se2159334234014336723it_int(Na,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_7726_flip__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Na)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_7727_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
         => ~ ! [Va2: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_7728_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_7729_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_7730_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_7731_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_7732_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_7733_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_7734_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_7735_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_7736_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_7737_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_7738_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se2159334234014336723it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_7739_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se2159334234014336723it_int(Na,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_7740_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_7741_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_en) = zero_zero_complex ).

% suminf_zero
tff(fact_7742_suminf__zero,axiom,
    suminf_real(aTP_Lamm_eo) = zero_zero_real ).

% suminf_zero
tff(fact_7743_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_cb) = zero_zero_nat ).

% suminf_zero
tff(fact_7744_suminf__zero,axiom,
    suminf_int(aTP_Lamm_ep) = zero_zero_int ).

% suminf_zero
tff(fact_7745_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_7746_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_7747_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_7748_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_7749_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_7750_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_7751_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_7752_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_7753_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_7754_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_7755_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_7756_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_7757_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_7758_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_7759_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_7760_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_7761_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(times_times_real(C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_7762_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_7763_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_7764_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_7765_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_7766_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_7767_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_7768_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_7769_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_7770_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_7771_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_7772_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_7773_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_7774_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_7775_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_7776_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_7777_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_7778_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_7779_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_diff
tff(fact_7780_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_7781_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_7782_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_7783_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_7784_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_7785_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_7786_norm__numeral,axiom,
    ! [W: num] : real_V1022390504157884413omplex(numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_7787_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_7788_norm__of__nat,axiom,
    ! [Na: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% norm_of_nat
tff(fact_7789_norm__of__nat,axiom,
    ! [Na: nat] : real_V1022390504157884413omplex(aa_nat_complex(semiri8010041392384452111omplex,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% norm_of_nat
tff(fact_7790_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_7791_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_7792_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_7793_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_7794_pow__divides__pow__iff,axiom,
    ! [Na: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_7795_pow__divides__pow__iff,axiom,
    ! [Na: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_7796_zero__less__norm__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xa))
    <=> ( Xa != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_7797_zero__less__norm__iff,axiom,
    ! [Xa: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),real_V1022390504157884413omplex(Xa))
    <=> ( Xa != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_7798_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_7799_norm__neg__numeral,axiom,
    ! [W: num] : real_V1022390504157884413omplex(uminus1482373934393186551omplex(numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7800_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_7801_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_7802_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_7803_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_7804_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_7805_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_7806_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_7807_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_7808_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),times_3573771949741848930nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_7809_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_7810_even__mod__2__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_7811_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_7812_even__Suc,axiom,
    ! [Na: 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,Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc
tff(fact_7813_even__Suc__Suc__iff,axiom,
    ! [Na: 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,Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc_Suc_iff
tff(fact_7814_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_7815_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),real_V1022390504157884413omplex(A)) ).

% norm_mult_numeral1
tff(fact_7816_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7817_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(times_times_real(real_V1022390504157884413omplex(A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7818_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7819_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : real_V1022390504157884413omplex(divide1717551699836669952omplex(A,numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(real_V1022390504157884413omplex(A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7820_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_fn(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7821_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_fo(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7822_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_7823_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7824_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_7825_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_7826_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_7827_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ).

% even_diff
tff(fact_7828_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).

% even_diff
tff(fact_7829_power__minus__odd,axiom,
    ! [Na: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Na) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% power_minus_odd
tff(fact_7830_power__minus__odd,axiom,
    ! [Na: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Na)) ) ) ).

% power_minus_odd
tff(fact_7831_power__minus__odd,axiom,
    ! [Na: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Na) = uminus1351360451143612070nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ) ).

% power_minus_odd
tff(fact_7832_power__minus__odd,axiom,
    ! [Na: nat,A: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(A)),Na) = uminus1482373934393186551omplex(aa_nat_complex(power_power_complex(A),Na)) ) ) ).

% power_minus_odd
tff(fact_7833_power__minus__odd,axiom,
    ! [Na: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Na) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% power_minus_odd
tff(fact_7834_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7835_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na) = aa_nat_int(power_power_int(A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7836_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Na) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7837_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(A)),Na) = aa_nat_complex(power_power_complex(A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7838_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Na) = aa_nat_rat(power_power_rat(A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7839_odd__Suc__div__two,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_7840_even__Suc__div__two,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_7841_power__even__abs__numeral,axiom,
    ! [W: num,A: 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(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7842_power__even__abs__numeral,axiom,
    ! [W: num,A: 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,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7843_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7844_power__even__abs__numeral,axiom,
    ! [W: num,A: 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,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7845_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_7846_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_7847_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_7848_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_7849_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_7850_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_7851_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_7852_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_7853_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_7854_even__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_7855_even__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),Na))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_7856_even__power,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_7857_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_7858_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,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_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_7859_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_7860_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_7861_power__less__zero__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),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))),Na)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_7862_power__less__zero__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),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))),Na)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_7863_power__less__zero__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),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))),Na)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_7864_power__less__zero__eq,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq
tff(fact_7865_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_7866_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_7867_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_7868_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq_numeral
tff(fact_7869_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_7870_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_7871_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Na) = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_7872_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na) = uminus1482373934393186551omplex(one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_7873_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_7874_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_7875_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_7876_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Na) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_7877_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_7878_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_7879_even__of__nat,axiom,
    ! [Na: 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,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_of_nat
tff(fact_7880_even__of__nat,axiom,
    ! [Na: 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,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_of_nat
tff(fact_7881_even__of__nat,axiom,
    ! [Na: 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,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_of_nat
tff(fact_7882_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_7883_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ) ) ).

% even_diff_nat
tff(fact_7884_norm__of__real__add1,axiom,
    ! [Xa: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xa)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),one_one_real)) ).

% norm_of_real_add1
tff(fact_7885_norm__of__real__add1,axiom,
    ! [Xa: real] : real_V1022390504157884413omplex(plus_plus_complex(real_V4546457046886955230omplex(Xa),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),one_one_real)) ).

% norm_of_real_add1
tff(fact_7886_norm__of__real__addn,axiom,
    ! [Xa: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xa)),aa_num_real(numeral_numeral_real,B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_7887_norm__of__real__addn,axiom,
    ! [Xa: real,B: num] : real_V1022390504157884413omplex(plus_plus_complex(real_V4546457046886955230omplex(Xa),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_7888_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( plus_p5714425477246183910nteger(times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_7889_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_7890_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_7891_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: 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))),Na)),one_one_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_7892_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_7893_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: 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))),Na),one_one_Code_integer))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_7894_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_7895_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_7896_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_7897_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_7898_odd__two__times__div__two__nat,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( 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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_7899_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_7900_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger)
    <=> ( 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_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_7901_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_7902_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_7903_even__succ__div__exp,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_7904_even__succ__div__exp,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_7905_even__succ__div__exp,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_7906_even__succ__mod__exp,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_7907_even__succ__mod__exp,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_7908_even__succ__mod__exp,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_7909_suminf__of__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => ( aa_real_real(real_V1803761363581548252l_real,suminf_real(X7)) = suminf_real(aTP_Lamm_fm(X7)) ) ) ).

% suminf_of_real
tff(fact_7910_suminf__of__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => ( real_V4546457046886955230omplex(suminf_real(X7)) = suminf_complex(aTP_Lamm_eq(X7)) ) ) ).

% suminf_of_real
tff(fact_7911_summable__norm,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fw(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_fw(F))) ) ).

% summable_norm
tff(fact_7912_summable__norm,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_fx(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(suminf_complex(F))),suminf_real(aTP_Lamm_fx(F))) ) ).

% summable_norm
tff(fact_7913_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(A),B)
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_7914_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_7915_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_7916_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_7917_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_7918_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_7919_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_7920_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_7921_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_7922_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_7923_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_7924_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ) ).

% dvd_add
tff(fact_7925_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ) ).

% dvd_add
tff(fact_7926_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_7927_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).

% dvd_add
tff(fact_7928_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(B),A)) ).

% dvd_triv_right
tff(fact_7929_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)) ).

% dvd_triv_right
tff(fact_7930_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_7931_dvd__triv__right,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) ).

% dvd_triv_right
tff(fact_7932_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_7933_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_7934_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_7935_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_7936_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_7937_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_7938_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_7939_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_7940_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(A),B)) ).

% dvd_triv_left
tff(fact_7941_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% dvd_triv_left
tff(fact_7942_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_7943_dvd__triv__left,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% dvd_triv_left
tff(fact_7944_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_7945_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_7946_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_7947_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_7948_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(B),C)) ) ).

% dvd_mult2
tff(fact_7949_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult2
tff(fact_7950_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_7951_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_7952_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(B),C)) ) ).

% dvd_mult
tff(fact_7953_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult
tff(fact_7954_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_7955_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_7956_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K3: real] : A = aa_real_real(times_times_real(B),K3) ) ).

% dvd_def
tff(fact_7957_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K3: rat] : A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K3) ) ).

% dvd_def
tff(fact_7958_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_7959_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K3: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvd_def
tff(fact_7960_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(times_times_real(B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_7961_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_7962_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_7963_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_7964_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K2: real] : A != aa_real_real(times_times_real(B),K2) ) ).

% dvdE
tff(fact_7965_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K2: rat] : A != aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K2) ) ).

% dvdE
tff(fact_7966_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_7967_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvdE
tff(fact_7968_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B7: nat,C6: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B7),C6) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B7),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C6),C) ) ) ).

% division_decomp
tff(fact_7969_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ? [B7: int,C6: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B7),C6) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B7),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C6),C) ) ) ).

% division_decomp
tff(fact_7970_dvd__productE,axiom,
    ! [P3: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X: nat,Y2: nat] :
            ( ( P3 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y2) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y2),B) ) ) ) ).

% dvd_productE
tff(fact_7971_dvd__productE,axiom,
    ! [P3: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P3),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ~ ! [X: int,Y2: int] :
            ( ( P3 = aa_int_int(aa_int_fun_int_int(times_times_int,X),Y2) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y2),B) ) ) ) ).

% dvd_productE
tff(fact_7972_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% dvd_diff_commute
tff(fact_7973_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(C),A)
     => ( aa_complex_o(dvd_dvd_complex(C),B)
       => ( ( divide1717551699836669952omplex(A,C) = divide1717551699836669952omplex(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_7974_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_7975_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_7976_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_7977_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_7978_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( divide1717551699836669952omplex(A,C) = divide1717551699836669952omplex(B,C) )
     => ( aa_complex_o(dvd_dvd_complex(C),A)
       => ( aa_complex_o(dvd_dvd_complex(C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_7979_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_7980_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_7981_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_7982_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_7983_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_7984_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_div_div_same
tff(fact_7985_dvd__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)
       => ( M = Na ) ) ) ).

% dvd_antisym
tff(fact_7986_of__nat__dvd__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% of_nat_dvd_iff
tff(fact_7987_of__nat__dvd__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% of_nat_dvd_iff
tff(fact_7988_of__nat__dvd__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% of_nat_dvd_iff
tff(fact_7989_dvd__power__same,axiom,
    ! [Xa: nat,Y: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Na)) ) ).

% dvd_power_same
tff(fact_7990_dvd__power__same,axiom,
    ! [Xa: int,Y: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xa),Na)),aa_nat_int(power_power_int(Y),Na)) ) ).

% dvd_power_same
tff(fact_7991_dvd__power__same,axiom,
    ! [Xa: real,Y: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na)) ) ).

% dvd_power_same
tff(fact_7992_dvd__power__same,axiom,
    ! [Xa: complex,Y: complex,Na: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xa),Y)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xa),Na)),aa_nat_complex(power_power_complex(Y),Na)) ) ).

% dvd_power_same
tff(fact_7993_dvd__power__same,axiom,
    ! [Xa: code_integer,Y: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Na)) ) ).

% dvd_power_same
tff(fact_7994_dvd__power__same,axiom,
    ! [Xa: rat,Y: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(Xa),Na)),aa_nat_rat(power_power_rat(Y),Na)) ) ).

% dvd_power_same
tff(fact_7995_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_7996_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_7997_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_7998_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_7999_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_8000_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_8001_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_8002_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) ) ) ).

% mod_mod_cancel
tff(fact_8003_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_8004_dvd__mod,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) ) ) ).

% dvd_mod
tff(fact_8005_dvd__mod,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),Na)) ) ) ).

% dvd_mod
tff(fact_8006_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,Na: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Na)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),modulo364778990260209775nteger(M,Na)) ) ) ).

% dvd_mod
tff(fact_8007_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_8008_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aTP_Lamm_fy(A))),collect_complex(aTP_Lamm_fy(B)))
    <=> aa_complex_o(dvd_dvd_complex(A),B) ) ).

% subset_divisors_dvd
tff(fact_8009_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_fz,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_fz,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_8010_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ga,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ga,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_8011_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( aa_set_complex_o(ord_less_set_complex(collect_complex(aTP_Lamm_fy(A))),collect_complex(aTP_Lamm_fy(B)))
    <=> ( aa_complex_o(dvd_dvd_complex(A),B)
        & ~ aa_complex_o(dvd_dvd_complex(B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8012_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ga,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ga,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8013_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_fz,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_fz,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8014_minf_I10_J,axiom,
    ! [D: real,Sa: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),Sa))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),Sa)) ) ) ).

% minf(10)
tff(fact_8015_minf_I10_J,axiom,
    ! [D: rat,Sa: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Sa))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Sa)) ) ) ).

% minf(10)
tff(fact_8016_minf_I10_J,axiom,
    ! [D: nat,Sa: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Sa))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Sa)) ) ) ).

% minf(10)
tff(fact_8017_minf_I10_J,axiom,
    ! [D: int,Sa: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Sa))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Sa)) ) ) ).

% minf(10)
tff(fact_8018_minf_I10_J,axiom,
    ! [D: code_integer,Sa: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X5,Sa))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X5,Sa)) ) ) ).

% minf(10)
tff(fact_8019_minf_I9_J,axiom,
    ! [D: real,Sa: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Z3)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),Sa))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),Sa)) ) ) ).

% minf(9)
tff(fact_8020_minf_I9_J,axiom,
    ! [D: rat,Sa: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Sa))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Sa)) ) ) ).

% minf(9)
tff(fact_8021_minf_I9_J,axiom,
    ! [D: nat,Sa: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Sa))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Sa)) ) ) ).

% minf(9)
tff(fact_8022_minf_I9_J,axiom,
    ! [D: int,Sa: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Sa))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Sa)) ) ) ).

% minf(9)
tff(fact_8023_minf_I9_J,axiom,
    ! [D: code_integer,Sa: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Z3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X5,Sa))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X5,Sa)) ) ) ).

% minf(9)
tff(fact_8024_pinf_I10_J,axiom,
    ! [D: real,Sa: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),Sa))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),Sa)) ) ) ).

% pinf(10)
tff(fact_8025_pinf_I10_J,axiom,
    ! [D: rat,Sa: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Sa))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Sa)) ) ) ).

% pinf(10)
tff(fact_8026_pinf_I10_J,axiom,
    ! [D: nat,Sa: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Sa))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Sa)) ) ) ).

% pinf(10)
tff(fact_8027_pinf_I10_J,axiom,
    ! [D: int,Sa: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Sa))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Sa)) ) ) ).

% pinf(10)
tff(fact_8028_pinf_I10_J,axiom,
    ! [D: code_integer,Sa: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X5,Sa))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X5,Sa)) ) ) ).

% pinf(10)
tff(fact_8029_pinf_I9_J,axiom,
    ! [D: real,Sa: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X5)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),Sa))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),Sa)) ) ) ).

% pinf(9)
tff(fact_8030_pinf_I9_J,axiom,
    ! [D: rat,Sa: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X5)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Sa))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),Sa)) ) ) ).

% pinf(9)
tff(fact_8031_pinf_I9_J,axiom,
    ! [D: nat,Sa: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X5)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Sa))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Sa)) ) ) ).

% pinf(9)
tff(fact_8032_pinf_I9_J,axiom,
    ! [D: int,Sa: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X5)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Sa))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Sa)) ) ) ).

% pinf(9)
tff(fact_8033_pinf_I9_J,axiom,
    ! [D: code_integer,Sa: code_integer] :
    ? [Z3: code_integer] :
    ! [X5: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X5,Sa))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X5,Sa)) ) ) ).

% pinf(9)
tff(fact_8034_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( ( divide1717551699836669952omplex(A,B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8035_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( ( aa_real_real(divide_divide_real(A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8036_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8037_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8038_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8039_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_8040_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_8041_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_8042_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_8043_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_8044_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_8045_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_8046_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_8047_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_8048_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_8049_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_8050_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_8051_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_8052_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_8053_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_8054_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_8055_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_8056_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_8057_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_8058_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_8059_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_8060_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_8061_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_8062_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_8063_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_8064_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_8065_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_8066_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_8067_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_8068_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_8069_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_8070_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_8071_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_8072_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_8073_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_8074_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_8075_norm__mult,axiom,
    ! [Xa: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Xa),Y)) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Y)) ).

% norm_mult
tff(fact_8076_norm__mult,axiom,
    ! [Xa: complex,Y: complex] : real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(Xa),Y)) = aa_real_real(times_times_real(real_V1022390504157884413omplex(Xa)),real_V1022390504157884413omplex(Y)) ).

% norm_mult
tff(fact_8077_norm__not__less__zero,axiom,
    ! [Xa: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(Xa)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_8078_norm__divide,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ).

% norm_divide
tff(fact_8079_norm__divide,axiom,
    ! [A: complex,B: complex] : real_V1022390504157884413omplex(divide1717551699836669952omplex(A,B)) = aa_real_real(divide_divide_real(real_V1022390504157884413omplex(A)),real_V1022390504157884413omplex(B)) ).

% norm_divide
tff(fact_8080_div__power,axiom,
    ! [B: code_integer,A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Na) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na)) ) ) ).

% div_power
tff(fact_8081_div__power,axiom,
    ! [B: nat,A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ).

% div_power
tff(fact_8082_div__power,axiom,
    ! [B: int,A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Na) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% div_power
tff(fact_8083_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_neg_div
tff(fact_8084_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_neg_div
tff(fact_8085_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(uminus1351360451143612070nteger(A),B) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_8086_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( divide1717551699836669952omplex(uminus1482373934393186551omplex(A),B) = uminus1482373934393186551omplex(divide1717551699836669952omplex(A,B)) ) ) ).

% dvd_neg_div
tff(fact_8087_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) ) ) ).

% dvd_neg_div
tff(fact_8088_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_div_neg
tff(fact_8089_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_div_neg
tff(fact_8090_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(A,uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_8091_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( divide1717551699836669952omplex(A,uminus1482373934393186551omplex(B)) = uminus1482373934393186551omplex(divide1717551699836669952omplex(A,B)) ) ) ).

% dvd_div_neg
tff(fact_8092_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) ) ) ).

% dvd_div_neg
tff(fact_8093_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_8094_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_8095_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_8096_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_8097_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_8098_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_8099_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_8100_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_8101_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_8102_norm__power,axiom,
    ! [Xa: real,Na: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xa)),Na) ).

% norm_power
tff(fact_8103_norm__power,axiom,
    ! [Xa: complex,Na: nat] : real_V1022390504157884413omplex(aa_nat_complex(power_power_complex(Xa),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Xa)),Na) ).

% norm_power
tff(fact_8104_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8105_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8106_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8107_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8108_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8109_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8110_power__le__dvd,axiom,
    ! [A: nat,Na: nat,B: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_8111_power__le__dvd,axiom,
    ! [A: int,Na: nat,B: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_8112_power__le__dvd,axiom,
    ! [A: real,Na: nat,B: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_8113_power__le__dvd,axiom,
    ! [A: complex,Na: nat,B: complex,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_8114_power__le__dvd,axiom,
    ! [A: code_integer,Na: nat,B: code_integer,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_8115_power__le__dvd,axiom,
    ! [A: rat,Na: nat,B: rat,M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_8116_dvd__power__le,axiom,
    ! [Xa: nat,Y: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M)) ) ) ).

% dvd_power_le
tff(fact_8117_dvd__power__le,axiom,
    ! [Xa: int,Y: int,Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xa),Na)),aa_nat_int(power_power_int(Y),M)) ) ) ).

% dvd_power_le
tff(fact_8118_dvd__power__le,axiom,
    ! [Xa: real,Y: real,Na: nat,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M)) ) ) ).

% dvd_power_le
tff(fact_8119_dvd__power__le,axiom,
    ! [Xa: complex,Y: complex,Na: nat,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xa),Na)),aa_nat_complex(power_power_complex(Y),M)) ) ) ).

% dvd_power_le
tff(fact_8120_dvd__power__le,axiom,
    ! [Xa: code_integer,Y: code_integer,Na: nat,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M)) ) ) ).

% dvd_power_le
tff(fact_8121_dvd__power__le,axiom,
    ! [Xa: rat,Y: rat,Na: nat,M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(Xa),Na)),aa_nat_rat(power_power_rat(Y),M)) ) ) ).

% dvd_power_le
tff(fact_8122_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ).

% dvd_minus_mod
tff(fact_8123_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% dvd_minus_mod
tff(fact_8124_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B))) ).

% dvd_minus_mod
tff(fact_8125_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ).

% mod_eq_dvd_iff
tff(fact_8126_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B)) ) ).

% mod_eq_dvd_iff
tff(fact_8127_dvd__pos__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_8128_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M) ) ) ).

% nat_dvd_not_less
tff(fact_8129_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_minus_self
tff(fact_8130_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_8131_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_8132_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_8133_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D2) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_8134_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y2)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_8135_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = D2 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_8136_suminf__add,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => ( plus_plus_complex(suminf_complex(F),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ex(F),G)) ) ) ) ).

% suminf_add
tff(fact_8137_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_ey(F),G)) ) ) ) ).

% suminf_add
tff(fact_8138_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_ez(F),G)) ) ) ) ).

% suminf_add
tff(fact_8139_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_fa(F),G)) ) ) ) ).

% suminf_add
tff(fact_8140_suminf__mult2,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( aa_complex_complex(times_times_complex(suminf_complex(F)),C) = suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fb(F),C)) ) ) ).

% suminf_mult2
tff(fact_8141_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_fc(F),C)) ) ) ).

% suminf_mult2
tff(fact_8142_suminf__mult,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fd(F),C)) = aa_complex_complex(times_times_complex(C),suminf_complex(F)) ) ) ).

% suminf_mult
tff(fact_8143_suminf__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_fe(F),C)) = aa_real_real(times_times_real(C),suminf_real(F)) ) ) ).

% suminf_mult
tff(fact_8144_suminf__diff,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => ( minus_minus_complex(suminf_complex(F),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ff(F),G)) ) ) ) ).

% suminf_diff
tff(fact_8145_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_fg(F),G)) ) ) ) ).

% suminf_diff
tff(fact_8146_suminf__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_eg(F),C)) = divide1717551699836669952omplex(suminf_complex(F),C) ) ) ).

% suminf_divide
tff(fact_8147_suminf__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_eh(F),C)) = aa_real_real(divide_divide_real(suminf_real(F)),C) ) ) ).

% suminf_divide
tff(fact_8148_suminf__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_fj(F)) = aa_real_real(uminus_uminus_real,suminf_real(F)) ) ) ).

% suminf_minus
tff(fact_8149_suminf__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => ( suminf_complex(aTP_Lamm_fk(F)) = uminus1482373934393186551omplex(suminf_complex(F)) ) ) ).

% suminf_minus
tff(fact_8150_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1345352211410354436nteger(M,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_8151_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se2159334234014336723it_int(M,A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_8152_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_8153_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))),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_8154_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_8155_suminf__exist__split,axiom,
    ! [R: real,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
     => ( summable_real(F)
       => ? [N8: nat] :
          ! [N9: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N9)
           => 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_el(F),N9)))),R) ) ) ) ).

% suminf_exist_split
tff(fact_8156_suminf__exist__split,axiom,
    ! [R: real,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
     => ( summable_complex(F)
       => ? [N8: nat] :
          ! [N9: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N9)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_em(F),N9)))),R) ) ) ) ).

% suminf_exist_split
tff(fact_8157_summable__norm__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fw(F))
     => summable_real(F) ) ).

% summable_norm_cancel
tff(fact_8158_summable__norm__cancel,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_fx(F))
     => summable_complex(F) ) ).

% summable_norm_cancel
tff(fact_8159_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_8160_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_8161_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_8162_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_8163_suminf__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(C)),one_one_real)
     => ( suminf_complex(power_power_complex(C)) = divide1717551699836669952omplex(one_one_complex,minus_minus_complex(one_one_complex,C)) ) ) ).

% suminf_geometric
tff(fact_8164_norm__uminus__minus,axiom,
    ! [Xa: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xa)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) ).

% norm_uminus_minus
tff(fact_8165_norm__uminus__minus,axiom,
    ! [Xa: complex,Y: complex] : real_V1022390504157884413omplex(minus_minus_complex(uminus1482373934393186551omplex(Xa),Y)) = real_V1022390504157884413omplex(plus_plus_complex(Xa,Y)) ).

% norm_uminus_minus
tff(fact_8166_power__eq__imp__eq__norm,axiom,
    ! [W: real,Na: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_8167_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Na: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),Na) = aa_nat_complex(power_power_complex(Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( real_V1022390504157884413omplex(W) = real_V1022390504157884413omplex(Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_8168_even__numeral,axiom,
    ! [Na: 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,Na))) ).

% even_numeral
tff(fact_8169_even__numeral,axiom,
    ! [Na: 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,Na))) ).

% even_numeral
tff(fact_8170_even__numeral,axiom,
    ! [Na: 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,Na))) ).

% even_numeral
tff(fact_8171_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X4: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X4))
    <=> ? [X4: complex] :
          ( aa_complex_o(dvd_dvd_complex(L),plus_plus_complex(X4,zero_zero_complex))
          & aa_complex_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_8172_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X4: real] : aa_real_o(P,aa_real_real(times_times_real(L),X4))
    <=> ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),zero_zero_real))
          & aa_real_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_8173_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X4))
    <=> ? [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),zero_zero_rat))
          & aa_rat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_8174_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X4: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X4))
    <=> ? [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),zero_zero_nat))
          & aa_nat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_8175_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X4: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X4))
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),zero_zero_int))
          & aa_int_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_8176_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C4: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C4) ) ) ).

% unit_dvdE
tff(fact_8177_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C4: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C4) ) ) ).

% unit_dvdE
tff(fact_8178_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_8179_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( real_V1022390504157884413omplex(divide1717551699836669952omplex(A,B)) = aa_real_real(divide_divide_real(real_V1022390504157884413omplex(A)),real_V1022390504157884413omplex(B)) ) ) ).

% nonzero_norm_divide
tff(fact_8180_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_8181_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_8182_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_8183_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_8184_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_8185_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_8186_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_8187_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_8188_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_8189_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_8190_inf__period_I4_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D3)
     => ! [X5: real,K5: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),T))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(X5),aa_real_real(times_times_real(K5),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_8191_inf__period_I4_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D3)
     => ! [X5: rat,K5: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),T))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K5),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_8192_inf__period_I4_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X5: int,K5: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T))
        <=> ~ 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,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_8193_inf__period_I3_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D3)
     => ! [X5: real,K5: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),T))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(X5),aa_real_real(times_times_real(K5),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_8194_inf__period_I3_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D3)
     => ! [X5: rat,K5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),T))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K5),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_8195_inf__period_I3_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X5: int,K5: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T))
        <=> 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,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_8196_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_8197_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_8198_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_8199_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_8200_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_8201_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_8202_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_8203_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_8204_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_8205_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_8206_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_8207_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_8208_is__unit__power__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_8209_is__unit__power__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_8210_is__unit__power__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_8211_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_8212_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_8213_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_8214_norm__mult__less,axiom,
    ! [Xa: real,R: real,Y: real,Sa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xa)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),Sa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Xa),Y))),aa_real_real(times_times_real(R),Sa)) ) ) ).

% norm_mult_less
tff(fact_8215_norm__mult__less,axiom,
    ! [Xa: complex,R: real,Y: complex,Sa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(Xa)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(Y)),Sa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(Xa),Y))),aa_real_real(times_times_real(R),Sa)) ) ) ).

% norm_mult_less
tff(fact_8216_norm__mult__ineq,axiom,
    ! [Xa: real,Y: 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(Xa),Y))),aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_mult_ineq
tff(fact_8217_norm__mult__ineq,axiom,
    ! [Xa: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(Xa),Y))),aa_real_real(times_times_real(real_V1022390504157884413omplex(Xa)),real_V1022390504157884413omplex(Y))) ).

% norm_mult_ineq
tff(fact_8218_norm__triangle__lt,axiom,
    ! [Xa: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),E) ) ).

% norm_triangle_lt
tff(fact_8219_norm__triangle__lt,axiom,
    ! [Xa: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,real_V1022390504157884413omplex(Xa)),real_V1022390504157884413omplex(Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(plus_plus_complex(Xa,Y))),E) ) ).

% norm_triangle_lt
tff(fact_8220_norm__add__less,axiom,
    ! [Xa: real,R: real,Y: real,Sa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xa)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),Sa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),Sa)) ) ) ).

% norm_add_less
tff(fact_8221_norm__add__less,axiom,
    ! [Xa: complex,R: real,Y: complex,Sa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(Xa)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(Y)),Sa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(plus_plus_complex(Xa,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),Sa)) ) ) ).

% norm_add_less
tff(fact_8222_norm__triangle__le,axiom,
    ! [Xa: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),E) ) ).

% norm_triangle_le
tff(fact_8223_norm__triangle__le,axiom,
    ! [Xa: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,real_V1022390504157884413omplex(Xa)),real_V1022390504157884413omplex(Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(plus_plus_complex(Xa,Y))),E) ) ).

% norm_triangle_le
tff(fact_8224_norm__triangle__mono,axiom,
    ! [A: real,R: real,B: real,Sa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),Sa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),Sa)) ) ) ).

% norm_triangle_mono
tff(fact_8225_norm__triangle__mono,axiom,
    ! [A: complex,R: real,B: complex,Sa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(A)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(B)),Sa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(plus_plus_complex(A,B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),Sa)) ) ) ).

% norm_triangle_mono
tff(fact_8226_norm__triangle__ineq,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_triangle_ineq
tff(fact_8227_norm__triangle__ineq,axiom,
    ! [Xa: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(plus_plus_complex(Xa,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,real_V1022390504157884413omplex(Xa)),real_V1022390504157884413omplex(Y))) ).

% norm_triangle_ineq
tff(fact_8228_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_8229_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(plus_plus_complex(A,B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,real_V1022390504157884413omplex(A)),C)) ) ).

% norm_add_leD
tff(fact_8230_norm__diff__triangle__less,axiom,
    ! [Xa: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xa),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xa),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_8231_norm__diff__triangle__less,axiom,
    ! [Xa: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(minus_minus_complex(Xa,Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(minus_minus_complex(Y,Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(minus_minus_complex(Xa,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_8232_norm__power__ineq,axiom,
    ! [Xa: real,Na: 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,Xa),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xa)),Na)) ).

% norm_power_ineq
tff(fact_8233_norm__power__ineq,axiom,
    ! [Xa: complex,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(aa_nat_complex(power_power_complex(Xa),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Xa)),Na)) ).

% norm_power_ineq
tff(fact_8234_norm__diff__ineq,axiom,
    ! [A: 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,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ).

% norm_diff_ineq
tff(fact_8235_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(real_V1022390504157884413omplex(A)),real_V1022390504157884413omplex(B))),real_V1022390504157884413omplex(plus_plus_complex(A,B))) ).

% norm_diff_ineq
tff(fact_8236_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% dvd_imp_le
tff(fact_8237_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_mult_cancel
tff(fact_8238_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_8239_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X: nat,Y2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_8240_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_8241_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q2) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_8242_real__of__nat__div,axiom,
    ! [D: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Na)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),D)) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_8243_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fl(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_fl(F))) ) ).

% summable_rabs
tff(fact_8244_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_fx(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_8245_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))
        <=> ? [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I2)) ) ) ) ).

% suminf_pos_iff
tff(fact_8246_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))
        <=> ? [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I2)) ) ) ) ).

% suminf_pos_iff
tff(fact_8247_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))
        <=> ? [I2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I2)) ) ) ) ).

% suminf_pos_iff
tff(fact_8248_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_8249_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_8250_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_8251_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_8252_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_8253_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_8254_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_8255_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% odd_even_add
tff(fact_8256_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).

% odd_even_add
tff(fact_8257_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_8258_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_8259_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_8260_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B5: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B5) ) ).

% evenE
tff(fact_8261_evenE,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B5: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B5) ) ).

% evenE
tff(fact_8262_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ~ ! [B5: code_integer] : A != times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B5) ) ).

% evenE
tff(fact_8263_power__eq__1__iff,axiom,
    ! [W: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Na) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_8264_power__eq__1__iff,axiom,
    ! [W: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(W),Na) = one_one_complex )
     => ( ( real_V1022390504157884413omplex(W) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_8265_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) )
        & ( divide6298287555418463151nteger(A,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_8266_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_8267_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_8268_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_8269_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_8270_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_8271_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_8272_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B5: nat] :
              ( ( B5 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B5),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B5 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B5) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B5) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B5) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_8273_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B5: int] :
              ( ( B5 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B5),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B5 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B5) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B5) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B5) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_8274_even__minus,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_8275_even__minus,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_8276_odd__numeral,axiom,
    ! [Na: 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,Na))) ).

% odd_numeral
tff(fact_8277_odd__numeral,axiom,
    ! [Na: 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,Na))) ).

% odd_numeral
tff(fact_8278_odd__numeral,axiom,
    ! [Na: 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,Na))) ).

% odd_numeral
tff(fact_8279_dvd__power__iff,axiom,
    ! [Xa: nat,M: nat,Na: nat] :
      ( ( Xa != 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,Xa),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% dvd_power_iff
tff(fact_8280_dvd__power__iff,axiom,
    ! [Xa: int,M: nat,Na: nat] :
      ( ( Xa != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xa),M)),aa_nat_int(power_power_int(Xa),Na))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% dvd_power_iff
tff(fact_8281_dvd__power__iff,axiom,
    ! [Xa: code_integer,M: nat,Na: nat] :
      ( ( Xa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Na))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% dvd_power_iff
tff(fact_8282_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_8283_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(minus_minus_complex(plus_plus_complex(A,B),plus_plus_complex(C,D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,real_V1022390504157884413omplex(minus_minus_complex(A,C))),real_V1022390504157884413omplex(minus_minus_complex(B,D)))) ).

% norm_diff_triangle_ineq
tff(fact_8284_suminf__split__head,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => ( suminf_complex(aTP_Lamm_fi(F)) = minus_minus_complex(suminf_complex(F),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8285_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_fh(F)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8286_dvd__power,axiom,
    ! [Na: nat,Xa: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na)) ) ).

% dvd_power
tff(fact_8287_dvd__power,axiom,
    ! [Na: nat,Xa: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),aa_nat_int(power_power_int(Xa),Na)) ) ).

% dvd_power
tff(fact_8288_dvd__power,axiom,
    ! [Na: nat,Xa: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) ) ).

% dvd_power
tff(fact_8289_dvd__power,axiom,
    ! [Na: nat,Xa: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_complex ) )
     => aa_complex_o(dvd_dvd_complex(Xa),aa_nat_complex(power_power_complex(Xa),Na)) ) ).

% dvd_power
tff(fact_8290_dvd__power,axiom,
    ! [Na: nat,Xa: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Na)) ) ).

% dvd_power
tff(fact_8291_dvd__power,axiom,
    ! [Na: nat,Xa: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),aa_nat_rat(power_power_rat(Xa),Na)) ) ).

% dvd_power
tff(fact_8292_norm__less__p1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,Xa))),one_one_real))) ).

% norm_less_p1
tff(fact_8293_norm__less__p1,axiom,
    ! [Xa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(Xa)),real_V1022390504157884413omplex(plus_plus_complex(real_V4546457046886955230omplex(real_V1022390504157884413omplex(Xa)),one_one_complex))) ).

% norm_less_p1
tff(fact_8294_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(M,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_8295_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_ri631733984087533419it_int(M,A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_8296_even__even__mod__4__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
    <=> 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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_8297_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_8298_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_8299_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_8300_dvd__minus__add,axiom,
    ! [Q2: nat,Na: nat,R: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_8301_diff__mod__le,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),D)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,D),B)) ) ) ).

% diff_mod_le
tff(fact_8302_mod__nat__eqI,axiom,
    ! [R: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_8303_powser__inside,axiom,
    ! [F: fun_nat_real,Xa: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xa))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Z)) ) ) ).

% powser_inside
tff(fact_8304_powser__inside,axiom,
    ! [F: fun_nat_complex,Xa: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(Z)),real_V1022390504157884413omplex(Xa))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Z)) ) ) ).

% powser_inside
tff(fact_8305_complete__algebra__summable__geometric,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xa)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,Xa)) ) ).

% complete_algebra_summable_geometric
tff(fact_8306_complete__algebra__summable__geometric,axiom,
    ! [Xa: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(Xa)),one_one_real)
     => summable_complex(power_power_complex(Xa)) ) ).

% complete_algebra_summable_geometric
tff(fact_8307_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_8308_summable__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(C)),one_one_real)
     => summable_complex(power_power_complex(C)) ) ).

% summable_geometric
tff(fact_8309_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_8310_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_8311_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_8312_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_8313_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_8314_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_8315_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_8316_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_8317_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_8318_power__mono__odd,axiom,
    ! [Na: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ).

% power_mono_odd
tff(fact_8319_power__mono__odd,axiom,
    ! [Na: nat,A: code_integer,B: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na)) ) ) ).

% power_mono_odd
tff(fact_8320_power__mono__odd,axiom,
    ! [Na: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono_odd
tff(fact_8321_power__mono__odd,axiom,
    ! [Na: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono_odd
tff(fact_8322_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% odd_pos
tff(fact_8323_uminus__power__if,axiom,
    ! [A: real,Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))) ).

% uminus_power_if
tff(fact_8324_uminus__power__if,axiom,
    ! [A: int,Na: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_int(power_power_int(A),Na),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Na))) ).

% uminus_power_if
tff(fact_8325_uminus__power__if,axiom,
    ! [A: code_integer,Na: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na),uminus1351360451143612070nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))) ).

% uminus_power_if
tff(fact_8326_uminus__power__if,axiom,
    ! [A: complex,Na: nat] :
      aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_complex(power_power_complex(A),Na),uminus1482373934393186551omplex(aa_nat_complex(power_power_complex(A),Na))) ).

% uminus_power_if
tff(fact_8327_uminus__power__if,axiom,
    ! [A: rat,Na: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_rat(power_power_rat(A),Na),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Na))) ).

% uminus_power_if
tff(fact_8328_power__even__abs,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na) = aa_nat_int(power_power_int(A),Na) ) ) ).

% power_even_abs
tff(fact_8329_power__even__abs,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) ) ) ).

% power_even_abs
tff(fact_8330_power__even__abs,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) ) ) ).

% power_even_abs
tff(fact_8331_power__even__abs,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Na) = aa_nat_rat(power_power_rat(A),Na) ) ) ).

% power_even_abs
tff(fact_8332_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_8333_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se8260200283734997820nteger(M,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_8334_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se4203085406695923979it_int(M,A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_8335_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_8336_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se2793503036327961859nteger(M,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_8337_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se7879613467334960850it_int(M,A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_8338_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_8339_powser__insidea,axiom,
    ! [F: fun_nat_real,Xa: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xa))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_gb(F),Z)) ) ) ).

% powser_insidea
tff(fact_8340_powser__insidea,axiom,
    ! [F: fun_nat_complex,Xa: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,real_V1022390504157884413omplex(Z)),real_V1022390504157884413omplex(Xa))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_gc(F),Z)) ) ) ).

% powser_insidea
tff(fact_8341_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Z)) = plus_plus_complex(aa_nat_complex(F,zero_zero_nat),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fq(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8342_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Z)) = aa_real_real(aa_rea7340605660300375871l_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_fs(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8343_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fq(F),Z))),Z) = minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(F),Z)),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8344_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Z))
     => ( aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_fs(F),Z))),Z) = aa_real_real(minus_minus_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8345_square__norm__one,axiom,
    ! [Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,Xa) = one_one_real ) ) ).

% square_norm_one
tff(fact_8346_square__norm__one,axiom,
    ! [Xa: complex] :
      ( ( aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( real_V1022390504157884413omplex(Xa) = one_one_real ) ) ).

% square_norm_one
tff(fact_8347_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B5: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B5)),one_one_nat) ) ).

% oddE
tff(fact_8348_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B5: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B5)),one_one_int) ) ).

% oddE
tff(fact_8349_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ~ ! [B5: code_integer] : A != plus_p5714425477246183910nteger(times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),B5),one_one_Code_integer) ) ).

% oddE
tff(fact_8350_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_8351_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_8352_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_8353_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_8354_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_8355_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
         => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_8356_zero__le__power__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8357_zero__le__power__eq,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8358_zero__le__power__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8359_zero__le__power__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8360_zero__le__odd__power,axiom,
    ! [Na: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_8361_zero__le__odd__power,axiom,
    ! [Na: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ).

% zero_le_odd_power
tff(fact_8362_zero__le__odd__power,axiom,
    ! [Na: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_8363_zero__le__odd__power,axiom,
    ! [Na: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_8364_zero__le__even__power,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_le_even_power
tff(fact_8365_zero__le__even__power,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ).

% zero_le_even_power
tff(fact_8366_zero__le__even__power,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_le_even_power
tff(fact_8367_zero__le__even__power,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_le_even_power
tff(fact_8368_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_8369_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_8370_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_8371_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_complex,uminus1482373934393186551omplex(one_one_complex)) ).

% minus_one_power_iff
tff(fact_8372_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_8373_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),M)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_8374_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),M),aa_nat_complex(power_power_complex(W),M)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),real_V1022390504157884413omplex(minus_minus_complex(Z,W)))) ) ) ).

% norm_power_diff
tff(fact_8375_power__mono__even,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ).

% power_mono_even
tff(fact_8376_power__mono__even,axiom,
    ! [Na: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na)) ) ) ).

% power_mono_even
tff(fact_8377_power__mono__even,axiom,
    ! [Na: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono_even
tff(fact_8378_power__mono__even,axiom,
    ! [Na: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono_even
tff(fact_8379_Abel__lemma,axiom,
    ! [R: real,R0: real,A: fun_nat_complex,M7: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,R),R0)
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(real_V1022390504157884413omplex(aa_nat_complex(A,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N))),M7)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_gd(R),A)) ) ) ) ).

% Abel_lemma
tff(fact_8380_zero__less__power__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
    <=> ( ( Na = 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))),Na)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8381_zero__less__power__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
    <=> ( ( Na = 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))),Na)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8382_zero__less__power__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
    <=> ( ( Na = 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))),Na)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8383_zero__less__power__eq,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))
    <=> ( ( Na = 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))),Na)
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8384_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_8385_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,real_V1022390504157884413omplex(aa_nat_complex(F,aa_nat_nat(suc,N)))),aa_real_real(times_times_real(C),real_V1022390504157884413omplex(aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8386_even__mask__div__iff_H,axiom,
    ! [M: nat,Na: 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))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% even_mask_div_iff'
tff(fact_8387_even__mask__div__iff_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% even_mask_div_iff'
tff(fact_8388_even__mask__div__iff_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% even_mask_div_iff'
tff(fact_8389_power__le__zero__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8390_power__le__zero__eq,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8391_power__le__zero__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8392_power__le__zero__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8393_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),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,Na),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_8394_even__mask__div__iff,axiom,
    ! [M: nat,Na: 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))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% even_mask_div_iff
tff(fact_8395_even__mask__div__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% even_mask_div_iff
tff(fact_8396_even__mask__div__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% even_mask_div_iff
tff(fact_8397_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),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,Na),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_8398_Bernoulli__inequality__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_8399_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [Na: nat] :
      aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_8400_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8401_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8402_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8403_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_8404_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [Na: nat] :
      aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_8405_VEBT__internal_OTb_H_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_8406_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [Na: nat] :
      aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_8407_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Na: nat] :
      aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_8408_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va: nat] :
      aa_nat_nat(vEBT_V8346862874174094_d_u_p,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))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_8409_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va: nat] :
      aa_nat_nat(vEBT_V8646137997579335489_i_l_d,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))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_8410_VEBT__internal_OTb_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_8411_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_8412_div2__even__ext__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xa)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
       => ( Xa = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_8413_arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(arctan,Xa) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ef,Xa)) ) ) ).

% arctan_series
tff(fact_8414_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_ge) ).

% pi_series
tff(fact_8415_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,T: nat] :
      ( hoare_hoare_triple_o(P,C,Q)
     => ( aa_nat_o(time_TBOUND_o(C),T)
       => aa_nat_o(time_htt_o(P,C,Q),T) ) ) ).

% httI_TBOUND
tff(fact_8416_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( aa_nat_o(time_T8353473612707095248on_nat(C),T)
       => aa_nat_o(time_htt_option_nat(P,C,Q),T) ) ) ).

% httI_TBOUND
tff(fact_8417_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( aa_nat_o(time_T5737551269749752165_VEBTi(C),T)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,C,Q),T) ) ) ).

% httI_TBOUND
tff(fact_8418_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( aa_nat_o(time_TBOUND_nat(C),T)
       => aa_nat_o(time_htt_nat(P,C,Q),T) ) ) ).

% httI_TBOUND
tff(fact_8419_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q),T)
    <=> ( (B)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,F,Q),T) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_8420_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_o,Q: fun_o_assn,T: nat] :
      ( aa_nat_o(time_htt_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q),T)
    <=> ( (B)
       => aa_nat_o(time_htt_o(P,F,Q),T) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_8421_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( aa_nat_o(time_htt_option_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q),T)
    <=> ( (B)
       => aa_nat_o(time_htt_option_nat(P,F,Q),T) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_8422_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( aa_nat_o(time_htt_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B))),F,Q),T)
    <=> ( (B)
       => aa_nat_o(time_htt_nat(P,F,Q),T) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_8423_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_V3964819847710782039nserti(T,Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_insert(T,Xa))) ).

% vebt_inserti'_rf_abstr
tff(fact_8424_deleti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_V1365221501068881998eletei(T,Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_delete(T,Xa))) ) ).

% deleti'_rf_abstr
tff(fact_8425_builupicorr,axiom,
    ! [Na: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_vebt_buildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))) ).

% builupicorr
tff(fact_8426_builupi_Hcorr,axiom,
    ! [Na: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))) ).

% builupi'corr
tff(fact_8427_arctan__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(arctan,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% arctan_eq_zero_iff
tff(fact_8428_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_8429_zero__less__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% zero_less_arctan_iff
tff(fact_8430_arctan__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_8431_zero__le__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% zero_le_arctan_iff
tff(fact_8432_arctan__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_8433_int__dvd__int__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% int_dvd_int_iff
tff(fact_8434_zdvd1__eq,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),one_one_int)
    <=> ( aa_int_int(abs_abs_int,Xa) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_8435_nat__abs__dvd__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Na)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_abs_dvd_iff
tff(fact_8436_dvd__nat__abs__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_int_nat(nat2,aa_int_int(abs_abs_int,K)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ).

% dvd_nat_abs_iff
tff(fact_8437_arctan__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(arctan,Xa) = aa_real_real(arctan,Y) )
    <=> ( Xa = Y ) ) ).

% arctan_eq_iff
tff(fact_8438_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_8439_arctan__monotone,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone
tff(fact_8440_arctan__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% arctan_less_iff
tff(fact_8441_arctan__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% arctan_le_iff
tff(fact_8442_arctan__monotone_H,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone'
tff(fact_8443_zdvd__zdiffD,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),Na))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M) ) ) ).

% zdvd_zdiffD
tff(fact_8444_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).

% zdvd_antisym_abs
tff(fact_8445_arctan__minus,axiom,
    ! [Xa: real] : aa_real_real(arctan,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(arctan,Xa)) ).

% arctan_minus
tff(fact_8446_pi__not__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).

% pi_not_less_zero
tff(fact_8447_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_8448_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_8449_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_8450_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),Na)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M) ) ) ).

% zdvd_not_zless
tff(fact_8451_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),T)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),T)) ) ) ).

% zdvd_mono
tff(fact_8452_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Na))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na) ) ) ).

% zdvd_mult_cancel
tff(fact_8453_zdvd__reduce,axiom,
    ! [K: int,Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),aa_int_int(aa_int_fun_int_int(times_times_int,K),M)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na) ) ).

% zdvd_reduce
tff(fact_8454_zdvd__period,axiom,
    ! [A: int,D: int,Xa: int,T: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),T))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),T)) ) ) ).

% zdvd_period
tff(fact_8455_abs__div,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),Xa)
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,Xa)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_8456_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_8457_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_8458_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_8459_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_8460_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_8461_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_8462_real__of__int__div,axiom,
    ! [D: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Na)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),D)) = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_8463_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_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_8464_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_8465_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_8466_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_8467_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_8468_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_8469_zdvd__mult__cancel1,axiom,
    ! [M: int,Na: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),Na)),M)
      <=> ( aa_int_int(abs_abs_int,Na) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_8470_int__div__sub__1,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),M)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),one_one_int)),M) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),M)),one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),M)) ) ) ).

% int_div_sub_1
tff(fact_8471_aset_I10_J,axiom,
    ! [D: int,D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T))
           => ~ 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,X5),D3)),T)) ) ) ) ).

% aset(10)
tff(fact_8472_aset_I9_J,axiom,
    ! [D: int,D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T))
           => 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,X5),D3)),T)) ) ) ) ).

% aset(9)
tff(fact_8473_bset_I10_J,axiom,
    ! [D: int,D3: int,B2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T))
           => ~ 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,X5),D3)),T)) ) ) ) ).

% bset(10)
tff(fact_8474_bset_I9_J,axiom,
    ! [D: int,D3: int,B2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T))
           => 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,X5),D3)),T)) ) ) ) ).

% bset(9)
tff(fact_8475_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_8476_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_8477_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_8478_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_8479_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_8480_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_8481_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_8482_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_8483_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_8484_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_8485_vebt__buildupi__rule__basic,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot_set_nat)) ) ).

% vebt_buildupi_rule_basic
tff(fact_8486_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_8487_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,T: nat] :
      ( aa_nat_o(time_htt_o(P,C,Q),T)
     => hoare_hoare_triple_o(P,C,Q) ) ).

% htt_htD
tff(fact_8488_htt__htD,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( aa_nat_o(time_htt_option_nat(P,C,Q),T)
     => hoare_7629718768684598413on_nat(P,C,Q) ) ).

% htt_htD
tff(fact_8489_htt__htD,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(P,C,Q),T)
     => hoare_1429296392585015714_VEBTi(P,C,Q) ) ).

% htt_htD
tff(fact_8490_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( aa_nat_o(time_htt_nat(P,C,Q),T)
     => hoare_3067605981109127869le_nat(P,C,Q) ) ).

% htt_htD
tff(fact_8491_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_8492_eme1p,axiom,
    ! [Na: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na)),D) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),D)) ) ) ) ) ).

% eme1p
tff(fact_8493_emep1,axiom,
    ! [Na: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int)),D) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),D)),one_one_int) ) ) ) ) ).

% emep1
tff(fact_8494_htt__refine,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat,C7: heap_Time_Heap_nat] :
      ( aa_nat_o(time_htt_nat(P,C,Q),T)
     => ( refine1365783493865988805es_nat(C7,C)
       => aa_nat_o(time_htt_nat(P,C7,Q),T) ) ) ).

% htt_refine
tff(fact_8495_htt__refine,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat,C7: heap_T8145700208782473153_VEBTi] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(P,C,Q),T)
     => ( refine5565527176597971370_VEBTi(C7,C)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,C7,Q),T) ) ) ).

% htt_refine
tff(fact_8496_htt__refine,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,T: nat,C7: heap_Time_Heap_o] :
      ( aa_nat_o(time_htt_o(P,C,Q),T)
     => ( refine_Imp_refines_o(C7,C)
       => aa_nat_o(time_htt_o(P,C7,Q),T) ) ) ).

% htt_refine
tff(fact_8497_htt__refine,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat,C7: heap_T2636463487746394924on_nat] :
      ( aa_nat_o(time_htt_option_nat(P,C,Q),T)
     => ( refine7594492741263601813on_nat(C7,C)
       => aa_nat_o(time_htt_option_nat(P,C7,Q),T) ) ) ).

% htt_refine
tff(fact_8498_arctan__add,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(Xa),Y)))) ) ) ) ).

% arctan_add
tff(fact_8499_arctan__double,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,Xa)) = 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))),Xa)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_8500_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(B))),P),F,Q),T)
    <=> ( (B)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,F,Q),T) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_8501_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: fun_o_assn,T: nat] :
      ( aa_nat_o(time_htt_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(B))),P),F,Q),T)
    <=> ( (B)
       => aa_nat_o(time_htt_o(P,F,Q),T) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_8502_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( aa_nat_o(time_htt_option_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(B))),P),F,Q),T)
    <=> ( (B)
       => aa_nat_o(time_htt_option_nat(P,F,Q),T) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_8503_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( aa_nat_o(time_htt_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(B))),P),F,Q),T)
    <=> ( (B)
       => aa_nat_o(time_htt_nat(P,F,Q),T) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_8504_lowi__hT,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_o(time_htt_nat(one_one_assn,vEBT_VEBT_lowi(Xa,Na),aa_nat_fun_nat_assn(aTP_Lamm_gf(Xa),Na)),one_one_nat) ).

% lowi_hT
tff(fact_8505_highi__hT,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_o(time_htt_nat(one_one_assn,vEBT_VEBT_highi(Xa,Na),aa_nat_fun_nat_assn(aTP_Lamm_gg(Xa),Na)),one_one_nat) ).

% highi_hT
tff(fact_8506_sin__cos__npi,axiom,
    ! [Na: 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))),Na)))),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)),Na) ).

% sin_cos_npi
tff(fact_8507_lowi__h,axiom,
    ! [Xa: nat,Na: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_lowi(Xa,Na),aa_nat_fun_nat_assn(aTP_Lamm_gf(Xa),Na)) ).

% lowi_h
tff(fact_8508_highi__h,axiom,
    ! [Xa: nat,Na: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_highi(Xa,Na),aa_nat_fun_nat_assn(aTP_Lamm_gg(Xa),Na)) ).

% highi_h
tff(fact_8509_TBOUND__lowi,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_o(time_TBOUND_nat(vEBT_VEBT_lowi(Xa,Na)),one_one_nat) ).

% TBOUND_lowi
tff(fact_8510_TBOUND__highi,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_o(time_TBOUND_nat(vEBT_VEBT_highi(Xa,Na)),one_one_nat) ).

% TBOUND_highi
tff(fact_8511_sin__zero,axiom,
    sin_complex(zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_8512_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_8513_sin__minus,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xa)) ).

% sin_minus
tff(fact_8514_sin__minus,axiom,
    ! [Xa: complex] : sin_complex(uminus1482373934393186551omplex(Xa)) = uminus1482373934393186551omplex(sin_complex(Xa)) ).

% sin_minus
tff(fact_8515_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_8516_sin__pi__minus,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(pi),Xa)) = aa_real_real(sin_real,Xa) ).

% sin_pi_minus
tff(fact_8517_sin__of__real__pi,axiom,
    aa_real_real(sin_real,aa_real_real(real_V1803761363581548252l_real,pi)) = zero_zero_real ).

% sin_of_real_pi
tff(fact_8518_sin__of__real__pi,axiom,
    sin_complex(real_V4546457046886955230omplex(pi)) = zero_zero_complex ).

% sin_of_real_pi
tff(fact_8519_sin__periodic__pi2,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xa)) ).

% sin_periodic_pi2
tff(fact_8520_sin__periodic__pi,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xa)) ).

% sin_periodic_pi
tff(fact_8521_sin__minus__pi,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(Xa),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xa)) ).

% sin_minus_pi
tff(fact_8522_sin__npi,axiom,
    ! [Na: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_8523_sin__npi2,axiom,
    ! [Na: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Na))) = zero_zero_real ).

% sin_npi2
tff(fact_8524_sin__npi__int,axiom,
    ! [Na: int] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,Na))) = zero_zero_real ).

% sin_npi_int
tff(fact_8525_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_8526_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_8527_sin__periodic,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xa) ).

% sin_periodic
tff(fact_8528_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_8529_sin__of__real__pi__half,axiom,
    sin_complex(divide1717551699836669952omplex(real_V4546457046886955230omplex(pi),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_8530_sin__2npi,axiom,
    ! [Na: 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,Na))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_8531_sin__2pi__minus,axiom,
    ! [Xa: 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)),Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xa)) ).

% sin_2pi_minus
tff(fact_8532_sin__int__2pin,axiom,
    ! [Na: 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,Na))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8533_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_8534_sin__of__real,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(real_V1803761363581548252l_real,Xa)) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(sin_real,Xa)) ).

% sin_of_real
tff(fact_8535_sin__of__real,axiom,
    ! [Xa: real] : sin_complex(real_V4546457046886955230omplex(Xa)) = real_V4546457046886955230omplex(aa_real_real(sin_real,Xa)) ).

% sin_of_real
tff(fact_8536_sin__x__le__x,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),Xa) ) ).

% sin_x_le_x
tff(fact_8537_sin__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),one_one_real) ).

% sin_le_one
tff(fact_8538_abs__sin__x__le__abs__x,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xa))),aa_real_real(abs_abs_real,Xa)) ).

% abs_sin_x_le_abs_x
tff(fact_8539_sin__int__times__real,axiom,
    ! [M: int,Xa: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,M)),aa_real_real(real_V1803761363581548252l_real,Xa))) = 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,M)),Xa))) ).

% sin_int_times_real
tff(fact_8540_sin__int__times__real,axiom,
    ! [M: int,Xa: real] : sin_complex(aa_complex_complex(times_times_complex(ring_17405671764205052669omplex(M)),real_V4546457046886955230omplex(Xa))) = real_V4546457046886955230omplex(aa_real_real(sin_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,M)),Xa))) ).

% sin_int_times_real
tff(fact_8541_sin__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_gt_zero
tff(fact_8542_sin__x__ge__neg__x,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xa)),aa_real_real(sin_real,Xa)) ) ).

% sin_x_ge_neg_x
tff(fact_8543_sin__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_ge_zero
tff(fact_8544_sin__ge__minus__one,axiom,
    ! [Xa: 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,Xa)) ).

% sin_ge_minus_one
tff(fact_8545_abs__sin__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xa))),one_one_real) ).

% abs_sin_le_one
tff(fact_8546_sin__eq__0__pi,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),pi)
       => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
         => ( Xa = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8547_sin__zero__pi__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),pi)
     => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8548_sin__zero__iff__int2,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
    <=> ? [I2: int] : Xa = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I2)),pi) ) ).

% sin_zero_iff_int2
tff(fact_8549_sin__gt__zero__02,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) ) ) ).

% sin_gt_zero_02
tff(fact_8550_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != 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,Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8551_sin__gt__zero2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) ) ) ).

% sin_gt_zero2
tff(fact_8552_sin__lt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_8553_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_8554_sin__inj__pi,axiom,
    ! [Xa: real,Y: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xa) = aa_real_real(sin_real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_8555_sin__mono__le__eq,axiom,
    ! [Xa: real,Y: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),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,Xa)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_8556_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xa: 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))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Y)),aa_real_real(sin_real,Xa)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_8557_sin__le__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_8558_sin__less__zero,axiom,
    ! [Xa: 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)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xa)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_8559_sin__mono__less__eq,axiom,
    ! [Xa: real,Y: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),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,Xa)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8560_sin__monotone__2pi,axiom,
    ! [Y: real,Xa: 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))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Y)),aa_real_real(sin_real,Xa)) ) ) ) ).

% sin_monotone_2pi
tff(fact_8561_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X: 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))))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X) = Y )
            & ! [Y4: 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))))),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% sin_total
tff(fact_8562_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => 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,Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8563_sin__zero__iff__int,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
    <=> ? [I2: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I2)
          & ( Xa = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_8564_sin__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(sin_real,Xa) = 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)
            & ( Xa = 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_8565_sin__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
    <=> ( ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xa = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xa = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_8566_lowi__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_lowi(Xa,Na) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% lowi_def
tff(fact_8567_highi__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_highi(Xa,Na) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% highi_def
tff(fact_8568_sincos__total__2pi,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xa = aa_real_real(cos_real,T4) )
               => ( Y != aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8569_cos__pi__eq__zero,axiom,
    ! [M: 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))),M))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_8570_highsimp,axiom,
    ! [Xa: nat,Na: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Na)) = vEBT_VEBT_highi(Xa,Na) ).

% highsimp
tff(fact_8571_lowsimp,axiom,
    ! [Xa: nat,Na: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Na)) = vEBT_VEBT_lowi(Xa,Na) ).

% lowsimp
tff(fact_8572_cos__minus,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(cos_real,Xa) ).

% cos_minus
tff(fact_8573_cos__minus,axiom,
    ! [Xa: complex] : cos_complex(uminus1482373934393186551omplex(Xa)) = cos_complex(Xa) ).

% cos_minus
tff(fact_8574_cos__zero,axiom,
    cos_complex(zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_8575_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_8576_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_8577_cos__periodic__pi,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xa)) ).

% cos_periodic_pi
tff(fact_8578_cos__periodic__pi2,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xa)) ).

% cos_periodic_pi2
tff(fact_8579_cos__pi__minus,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(pi),Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xa)) ).

% cos_pi_minus
tff(fact_8580_cos__minus__pi,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(Xa),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xa)) ).

% cos_minus_pi
tff(fact_8581_sin__cos__squared__add3,axiom,
    ! [Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Xa))),aa_real_real(times_times_real(aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Xa))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_8582_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_8583_cos__of__real__pi,axiom,
    cos_complex(real_V4546457046886955230omplex(pi)) = uminus1482373934393186551omplex(one_one_complex) ).

% cos_of_real_pi
tff(fact_8584_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_8585_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_8586_cos__periodic,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xa) ).

% cos_periodic
tff(fact_8587_cos__2pi__minus,axiom,
    ! [Xa: 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)),Xa)) = aa_real_real(cos_real,Xa) ).

% cos_2pi_minus
tff(fact_8588_cos__npi2,axiom,
    ! [Na: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) ).

% cos_npi2
tff(fact_8589_cos__npi,axiom,
    ! [Na: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) ).

% cos_npi
tff(fact_8590_sin__cos__squared__add2,axiom,
    ! [Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_8591_sin__cos__squared__add2,axiom,
    ! [Xa: complex] : plus_plus_complex(aa_nat_complex(power_power_complex(cos_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(sin_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_8592_sin__cos__squared__add,axiom,
    ! [Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_8593_sin__cos__squared__add,axiom,
    ! [Xa: complex] : plus_plus_complex(aa_nat_complex(power_power_complex(sin_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(cos_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_8594_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_8595_cos__of__real__pi__half,axiom,
    cos_complex(divide1717551699836669952omplex(real_V4546457046886955230omplex(pi),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_8596_cos__2npi,axiom,
    ! [Na: 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,Na))),pi)) = one_one_real ).

% cos_2npi
tff(fact_8597_cos__int__2pin,axiom,
    ! [Na: 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,Na))) = one_one_real ).

% cos_int_2pin
tff(fact_8598_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_8599_cos__npi__int,axiom,
    ! [Na: int] :
      aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,Na))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_8600_cos__of__real,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(real_V1803761363581548252l_real,Xa)) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(cos_real,Xa)) ).

% cos_of_real
tff(fact_8601_cos__of__real,axiom,
    ! [Xa: real] : cos_complex(real_V4546457046886955230omplex(Xa)) = real_V4546457046886955230omplex(aa_real_real(cos_real,Xa)) ).

% cos_of_real
tff(fact_8602_cos__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),one_one_real) ).

% cos_le_one
tff(fact_8603_polar__Ex,axiom,
    ! [Xa: real,Y: real] :
    ? [R4: real,A4: real] :
      ( ( Xa = aa_real_real(times_times_real(R4),aa_real_real(cos_real,A4)) )
      & ( Y = aa_real_real(times_times_real(R4),aa_real_real(sin_real,A4)) ) ) ).

% polar_Ex
tff(fact_8604_cos__arctan__not__zero,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(arctan,Xa)) != zero_zero_real ).

% cos_arctan_not_zero
tff(fact_8605_cos__int__times__real,axiom,
    ! [M: int,Xa: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,M)),aa_real_real(real_V1803761363581548252l_real,Xa))) = 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,M)),Xa))) ).

% cos_int_times_real
tff(fact_8606_cos__int__times__real,axiom,
    ! [M: int,Xa: real] : cos_complex(aa_complex_complex(times_times_complex(ring_17405671764205052669omplex(M)),real_V4546457046886955230omplex(Xa))) = real_V4546457046886955230omplex(aa_real_real(cos_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,M)),Xa))) ).

% cos_int_times_real
tff(fact_8607_cos__one__sin__zero,axiom,
    ! [Xa: complex] :
      ( ( cos_complex(Xa) = one_one_complex )
     => ( sin_complex(Xa) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_8608_cos__one__sin__zero,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = one_one_real )
     => ( aa_real_real(sin_real,Xa) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_8609_sin__add,axiom,
    ! [Xa: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(sin_real,Xa)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xa)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_8610_sin__diff,axiom,
    ! [Xa: real,Y: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(Xa),Y)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(sin_real,Xa)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xa)),aa_real_real(sin_real,Y))) ).

% sin_diff
tff(fact_8611_cos__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( ( aa_real_real(cos_real,Xa) = aa_real_real(cos_real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_8612_cos__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_8613_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_8614_cos__ge__minus__one,axiom,
    ! [Xa: 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,Xa)) ).

% cos_ge_minus_one
tff(fact_8615_abs__cos__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xa))),one_one_real) ).

% abs_cos_le_one
tff(fact_8616_cos__add,axiom,
    ! [Xa: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_8617_cos__diff,axiom,
    ! [Xa: real,Y: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_8618_sin__zero__norm__cos__one,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,Xa)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_8619_sin__zero__norm__cos__one,axiom,
    ! [Xa: complex] :
      ( ( sin_complex(Xa) = zero_zero_complex )
     => ( real_V1022390504157884413omplex(cos_complex(Xa)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_8620_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_8621_cos__monotone__0__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8622_cos__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8623_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_8624_sin__zero__abs__cos__one,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xa)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_8625_sin__double,axiom,
    ! [Xa: complex] : sin_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),sin_complex(Xa))),cos_complex(Xa)) ).

% sin_double
tff(fact_8626_sin__double,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)) = 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,Xa))),aa_real_real(cos_real,Xa)) ).

% sin_double
tff(fact_8627_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_8628_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_8629_cos__is__zero,axiom,
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X) = zero_zero_real )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X ) ) ) ).

% cos_is_zero
tff(fact_8630_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8631_cos__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
            & ( aa_real_real(cos_real,X) = Y )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),pi)
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% cos_total
tff(fact_8632_sincos__principal__value,axiom,
    ! [Xa: real] :
    ? [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
      & ( aa_real_real(sin_real,Y2) = aa_real_real(sin_real,Xa) )
      & ( aa_real_real(cos_real,Y2) = aa_real_real(cos_real,Xa) ) ) ).

% sincos_principal_value
tff(fact_8633_sin__cos__le1,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))))),one_one_real) ).

% sin_cos_le1
tff(fact_8634_cos__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(cos_complex(W)),cos_complex(Z)) = divide1717551699836669952omplex(plus_plus_complex(cos_complex(minus_minus_complex(W,Z)),cos_complex(plus_plus_complex(W,Z))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_8635_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_8636_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] : plus_plus_complex(cos_complex(W),cos_complex(Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),cos_complex(divide1717551699836669952omplex(plus_plus_complex(W,Z),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),cos_complex(divide1717551699836669952omplex(minus_minus_complex(W,Z),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_8637_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_8638_sin__squared__eq,axiom,
    ! [Xa: complex] : aa_nat_complex(power_power_complex(sin_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_complex(one_one_complex,aa_nat_complex(power_power_complex(cos_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_8639_sin__squared__eq,axiom,
    ! [Xa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_8640_cos__squared__eq,axiom,
    ! [Xa: complex] : aa_nat_complex(power_power_complex(cos_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_complex(one_one_complex,aa_nat_complex(power_power_complex(sin_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_8641_cos__squared__eq,axiom,
    ! [Xa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_8642_cos__double__less__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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))),Xa))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_8643_cos__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) ) ) ).

% cos_gt_zero
tff(fact_8644_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_8645_cos__one__2pi__int,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = one_one_real )
    <=> ? [X4: int] : Xa = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_8646_cos__double__cos,axiom,
    ! [W: complex] : cos_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(cos_complex(W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% cos_double_cos
tff(fact_8647_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_8648_cos__treble__cos,axiom,
    ! [Xa: complex] : cos_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),Xa)) = minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(cos_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),cos_complex(Xa))) ).

% cos_treble_cos
tff(fact_8649_cos__treble__cos,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),Xa)) = 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,Xa)),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,Xa))) ).

% cos_treble_cos
tff(fact_8650_sin__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(sin_complex(W)),sin_complex(Z)) = divide1717551699836669952omplex(minus_minus_complex(cos_complex(minus_minus_complex(W,Z)),cos_complex(plus_plus_complex(W,Z))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_8651_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(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_8652_sin__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(sin_complex(W)),cos_complex(Z)) = divide1717551699836669952omplex(plus_plus_complex(sin_complex(plus_plus_complex(W,Z)),sin_complex(minus_minus_complex(W,Z))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_8653_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_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_8654_cos__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(cos_complex(W)),sin_complex(Z)) = divide1717551699836669952omplex(minus_minus_complex(sin_complex(plus_plus_complex(W,Z)),sin_complex(minus_minus_complex(W,Z))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_8655_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(aa_rea7340605660300375871l_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_8656_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] : plus_plus_complex(sin_complex(W),sin_complex(Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),sin_complex(divide1717551699836669952omplex(plus_plus_complex(W,Z),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),cos_complex(divide1717551699836669952omplex(minus_minus_complex(W,Z),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_8657_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_8658_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] : minus_minus_complex(sin_complex(W),sin_complex(Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),sin_complex(divide1717551699836669952omplex(minus_minus_complex(W,Z),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),cos_complex(divide1717551699836669952omplex(plus_plus_complex(W,Z),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_8659_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(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_8660_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] : minus_minus_complex(cos_complex(W),cos_complex(Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),sin_complex(divide1717551699836669952omplex(plus_plus_complex(W,Z),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),sin_complex(divide1717551699836669952omplex(minus_minus_complex(Z,W),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_8661_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(aa_rea7340605660300375871l_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_8662_cos__double,axiom,
    ! [Xa: complex] : cos_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)) = minus_minus_complex(aa_nat_complex(power_power_complex(cos_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(sin_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_8663_cos__double,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_8664_sin__cos__eq,axiom,
    ! [Xa: real] : aa_real_real(sin_real,Xa) = 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)))),Xa)) ).

% sin_cos_eq
tff(fact_8665_sin__cos__eq,axiom,
    ! [Xa: complex] : sin_complex(Xa) = cos_complex(minus_minus_complex(divide1717551699836669952omplex(real_V4546457046886955230omplex(pi),numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)) ).

% sin_cos_eq
tff(fact_8666_cos__sin__eq,axiom,
    ! [Xa: real] : aa_real_real(cos_real,Xa) = 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)))),Xa)) ).

% cos_sin_eq
tff(fact_8667_cos__sin__eq,axiom,
    ! [Xa: complex] : cos_complex(Xa) = sin_complex(minus_minus_complex(divide1717551699836669952omplex(real_V4546457046886955230omplex(pi),numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)) ).

% cos_sin_eq
tff(fact_8668_cos__gt__zero__pi,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) ) ) ).

% cos_gt_zero_pi
tff(fact_8669_cos__ge__zero,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)) ) ) ).

% cos_ge_zero
tff(fact_8670_cos__one__2pi,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = one_one_real )
    <=> ( ? [X4: nat] : Xa = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X4: nat] : Xa = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_8671_cos__double__sin,axiom,
    ! [W: complex] : cos_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = minus_minus_complex(one_one_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(sin_complex(W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_8672_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_8673_minus__sin__cos__eq,axiom,
    ! [Xa: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xa)) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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_8674_minus__sin__cos__eq,axiom,
    ! [Xa: complex] : uminus1482373934393186551omplex(sin_complex(Xa)) = cos_complex(plus_plus_complex(Xa,divide1717551699836669952omplex(real_V4546457046886955230omplex(pi),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_8675_sincos__total__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),pi)
            & ( Xa = aa_real_real(cos_real,T4) )
            & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_8676_sin__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_8677_cos__zero__iff__int,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = zero_zero_real )
    <=> ? [I2: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I2)
          & ( Xa = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_8678_cos__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(cos_real,Xa) = 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)
            & ( Xa = 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_8679_cos__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = zero_zero_real )
    <=> ( ? [N2: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xa = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N2: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xa = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_8680_cos__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_8681_sincos__total__pi__half,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xa = aa_real_real(cos_real,T4) )
              & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_8682_sincos__total__2pi__le,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xa = aa_real_real(cos_real,T4) )
          & ( Y = aa_real_real(sin_real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_8683_return__sp__rule,axiom,
    ! [P: assn,Xa: $o] : hoare_hoare_triple_o(P,heap_Time_return_o((Xa)),aa_o_fun_o_assn(aTP_Lamm_gh(P),(Xa))) ).

% return_sp_rule
tff(fact_8684_return__sp__rule,axiom,
    ! [P: assn,Xa: option_nat] : hoare_7629718768684598413on_nat(P,heap_T3487192422709364219on_nat(Xa),aa_opt3861602675778688675t_assn(aTP_Lamm_gi(P),Xa)) ).

% return_sp_rule
tff(fact_8685_return__sp__rule,axiom,
    ! [P: assn,Xa: vEBT_VEBTi] : hoare_1429296392585015714_VEBTi(P,heap_T3630416162098727440_VEBTi(Xa),aa_VEB1545114916007340237i_assn(aTP_Lamm_gj(P),Xa)) ).

% return_sp_rule
tff(fact_8686_return__sp__rule,axiom,
    ! [P: assn,Xa: nat] : hoare_3067605981109127869le_nat(P,heap_Time_return_nat(Xa),aa_nat_fun_nat_assn(aTP_Lamm_gk(P),Xa)) ).

% return_sp_rule
tff(fact_8687_tan__double,axiom,
    ! [Xa: complex] :
      ( ( cos_complex(Xa) != zero_zero_complex )
     => ( ( cos_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)) != zero_zero_complex )
       => ( tan_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)) = divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),tan_complex(Xa)),minus_minus_complex(one_one_complex,aa_nat_complex(power_power_complex(tan_complex(Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8688_tan__double,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) != 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))),Xa)) != 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))),Xa)) = 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,Xa))),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8689_triangle__def,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_8690_obtain__set__succ,axiom,
    ! [Xa: nat,Z: nat,A2: set_nat,B2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => ( ( A2 = B2 )
           => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A2),Xa),X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_8691_obtain__set__pred,axiom,
    ! [Z: nat,Xa: nat,A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xa)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A2),Xa),X_1) ) ) ) ).

% obtain_set_pred
tff(fact_8692_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(T)) ) ).

% set_vebt_finite
tff(fact_8693_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),A) ) ) ) ).

% pred_none_empty
tff(fact_8694_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X5) ) ) ) ).

% succ_none_empty
tff(fact_8695_false__rule,axiom,
    ! [C: heap_Time_Heap_o,Q: fun_o_assn] : hoare_hoare_triple_o(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_8696_false__rule,axiom,
    ! [C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] : hoare_7629718768684598413on_nat(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_8697_false__rule,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] : hoare_1429296392585015714_VEBTi(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_8698_false__rule,axiom,
    ! [C: heap_Time_Heap_nat,Q: fun_nat_assn] : hoare_3067605981109127869le_nat(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_8699_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_8700_tan__zero,axiom,
    tan_complex(zero_zero_complex) = zero_zero_complex ).

% tan_zero
tff(fact_8701_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_8702_tan__minus,axiom,
    ! [Xa: real] : aa_real_real(tan_real,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(tan_real,Xa)) ).

% tan_minus
tff(fact_8703_tan__minus,axiom,
    ! [Xa: complex] : tan_complex(uminus1482373934393186551omplex(Xa)) = uminus1482373934393186551omplex(tan_complex(Xa)) ).

% tan_minus
tff(fact_8704_tan__periodic__pi,axiom,
    ! [Xa: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),pi)) = aa_real_real(tan_real,Xa) ).

% tan_periodic_pi
tff(fact_8705_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Icc_iff
tff(fact_8706_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Icc_iff
tff(fact_8707_triangle__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Na)),aa_nat_nat(suc,Na)) ).

% triangle_Suc
tff(fact_8708_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_gl(P),F)) ) ).

% summable_If_finite
tff(fact_8709_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_gm(P),F)) ) ).

% summable_If_finite
tff(fact_8710_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_gn(P),F)) ) ).

% summable_If_finite
tff(fact_8711_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_go(P),F)) ) ).

% summable_If_finite
tff(fact_8712_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gp(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8713_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_gq(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8714_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gr(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8715_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_gs(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8716_Min_Obounded__iff,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),lattic3629708407755379051n_real(A2))
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X4) ) ) ) ) ).

% Min.bounded_iff
tff(fact_8717_Min_Obounded__iff,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic8086005427650270231in_rat(A2))
        <=> ! [X4: rat] :
              ( aa_set_rat_o(member_rat(X4),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X4) ) ) ) ) ).

% Min.bounded_iff
tff(fact_8718_Min_Obounded__iff,axiom,
    ! [A2: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),lattic5278467273892544601in_num(A2))
        <=> ! [X4: num] :
              ( aa_set_num_o(member_num(X4),A2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X4) ) ) ) ) ).

% Min.bounded_iff
tff(fact_8719_Min_Obounded__iff,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic8718645017227715691in_int(A2))
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X4) ) ) ) ) ).

% Min.bounded_iff
tff(fact_8720_Min_Obounded__iff,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),lattic8721135487736765967in_nat(A2))
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4) ) ) ) ) ).

% Min.bounded_iff
tff(fact_8721_Min__gr__iff,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),lattic3629708407755379051n_real(A2))
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X4) ) ) ) ) ).

% Min_gr_iff
tff(fact_8722_Min__gr__iff,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),lattic8086005427650270231in_rat(A2))
        <=> ! [X4: rat] :
              ( aa_set_rat_o(member_rat(X4),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X4) ) ) ) ) ).

% Min_gr_iff
tff(fact_8723_Min__gr__iff,axiom,
    ! [A2: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),lattic5278467273892544601in_num(A2))
        <=> ! [X4: num] :
              ( aa_set_num_o(member_num(X4),A2)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X4) ) ) ) ) ).

% Min_gr_iff
tff(fact_8724_Min__gr__iff,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),lattic8718645017227715691in_int(A2))
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X4) ) ) ) ) ).

% Min_gr_iff
tff(fact_8725_Min__gr__iff,axiom,
    ! [A2: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),lattic1063845414844153500nteger(A2))
        <=> ! [X4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),X4) ) ) ) ) ).

% Min_gr_iff
tff(fact_8726_Min__gr__iff,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),lattic8721135487736765967in_nat(A2))
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X4) ) ) ) ) ).

% Min_gr_iff
tff(fact_8727_tan__npi,axiom,
    ! [Na: nat] : aa_real_real(tan_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_8728_tan__periodic__n,axiom,
    ! [Xa: real,Na: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Na)),pi))) = aa_real_real(tan_real,Xa) ).

% tan_periodic_n
tff(fact_8729_tan__periodic__nat,axiom,
    ! [Xa: real,Na: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi))) = aa_real_real(tan_real,Xa) ).

% tan_periodic_nat
tff(fact_8730_tan__periodic__int,axiom,
    ! [Xa: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,Xa) ).

% tan_periodic_int
tff(fact_8731_tan__periodic,axiom,
    ! [Xa: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xa) ).

% tan_periodic
tff(fact_8732_tan__arctan,axiom,
    ! [Y: real] : aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ).

% tan_arctan
tff(fact_8733_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,Na: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Na) )
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% bounded_nat_set_is_finite
tff(fact_8734_finite__nat__set__iff__bounded,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_8735_finite__if__eq__beyond__finite,axiom,
    ! [S: set_int,S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_gt(S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8736_finite__if__eq__beyond__finite,axiom,
    ! [S: set_complex,S3: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => finite6551019134538273531omplex(collect_set_complex(aa_set8643480048548153067plex_o(aTP_Lamm_gu(S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8737_finite__if__eq__beyond__finite,axiom,
    ! [S: set_nat,S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => finite1152437895449049373et_nat(collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_gv(S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8738_tan__of__real,axiom,
    ! [Xa: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(tan_real,Xa)) = aa_real_real(tan_real,aa_real_real(real_V1803761363581548252l_real,Xa)) ).

% tan_of_real
tff(fact_8739_tan__of__real,axiom,
    ! [Xa: real] : real_V4546457046886955230omplex(aa_real_real(tan_real,Xa)) = tan_complex(real_V4546457046886955230omplex(Xa)) ).

% tan_of_real
tff(fact_8740_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_gw(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_8741_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_gx(F),U))) ) ).

% finite_less_ub
tff(fact_8742_infinite__growing,axiom,
    ! [X7: set_real] :
      ( ( X7 != bot_bot_set_real )
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),X7)
           => ? [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),X7)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X),Xa2) ) )
       => ~ aa_set_real_o(finite_finite_real,X7) ) ) ).

% infinite_growing
tff(fact_8743_infinite__growing,axiom,
    ! [X7: set_rat] :
      ( ( X7 != bot_bot_set_rat )
     => ( ! [X: rat] :
            ( aa_set_rat_o(member_rat(X),X7)
           => ? [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),X7)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Xa2) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X7) ) ) ).

% infinite_growing
tff(fact_8744_infinite__growing,axiom,
    ! [X7: set_num] :
      ( ( X7 != bot_bot_set_num )
     => ( ! [X: num] :
            ( aa_set_num_o(member_num(X),X7)
           => ? [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),X7)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X),Xa2) ) )
       => ~ aa_set_num_o(finite_finite_num,X7) ) ) ).

% infinite_growing
tff(fact_8745_infinite__growing,axiom,
    ! [X7: set_nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),X7)
           => ? [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),X7)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa2) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X7) ) ) ).

% infinite_growing
tff(fact_8746_infinite__growing,axiom,
    ! [X7: set_int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),X7)
           => ? [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),X7)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa2) ) )
       => ~ aa_set_int_o(finite_finite_int,X7) ) ) ).

% infinite_growing
tff(fact_8747_infinite__growing,axiom,
    ! [X7: set_Code_integer] :
      ( ( X7 != bot_bo3990330152332043303nteger )
     => ( ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),X7)
           => ? [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),X7)
                & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Xa2) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X7) ) ) ).

% infinite_growing
tff(fact_8748_ex__min__if__finite,axiom,
    ! [S: set_real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ( S != bot_bot_set_real )
       => ? [X: real] :
            ( aa_set_real_o(member_real(X),S)
            & ~ ? [Xa2: real] :
                  ( aa_set_real_o(member_real(Xa2),S)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8749_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S)
     => ( ( S != bot_bot_set_rat )
       => ? [X: rat] :
            ( aa_set_rat_o(member_rat(X),S)
            & ~ ? [Xa2: rat] :
                  ( aa_set_rat_o(member_rat(Xa2),S)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8750_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( aa_set_num_o(finite_finite_num,S)
     => ( ( S != bot_bot_set_num )
       => ? [X: num] :
            ( aa_set_num_o(member_num(X),S)
            & ~ ? [Xa2: num] :
                  ( aa_set_num_o(member_num(Xa2),S)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8751_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),S)
            & ~ ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),S)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8752_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),S)
            & ~ ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),S)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8753_ex__min__if__finite,axiom,
    ! [S: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ? [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),S)
            & ~ ? [Xa2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),S)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8754_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_8755_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_8756_Min__in,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => aa_set_real_o(member_real(lattic3629708407755379051n_real(A2)),A2) ) ) ).

% Min_in
tff(fact_8757_Min__in,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => aa_set_int_o(member_int(lattic8718645017227715691in_int(A2)),A2) ) ) ).

% Min_in
tff(fact_8758_Min__in,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(lattic8721135487736765967in_nat(A2)),A2) ) ) ).

% Min_in
tff(fact_8759_finite__ranking__induct,axiom,
    ! [S: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X: vEBT_VEBT,S4: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
             => ( ! [Y4: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,Y4)),aa_VEBT_VEBT_rat(F,X)) )
               => ( aa_set_VEBT_VEBT_o(P,S4)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X,S4)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8760_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_o,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X: complex,S4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S4)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Y4)),aa_complex_rat(F,X)) )
               => ( aa_set_complex_o(P,S4)
                 => aa_set_complex_o(P,insert_complex(X,S4)) ) ) )
         => aa_set_complex_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8761_finite__ranking__induct,axiom,
    ! [S: set_real,P: fun_set_real_o,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X: real,S4: set_real] :
              ( aa_set_real_o(finite_finite_real,S4)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Y4)),aa_real_rat(F,X)) )
               => ( aa_set_real_o(P,S4)
                 => aa_set_real_o(P,insert_real(X,S4)) ) ) )
         => aa_set_real_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8762_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X: nat,S4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S4)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X)) )
               => ( aa_set_nat_o(P,S4)
                 => aa_set_nat_o(P,insert_nat(X,S4)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8763_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X: int,S4: set_int] :
              ( aa_set_int_o(finite_finite_int,S4)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X)) )
               => ( aa_set_int_o(P,S4)
                 => aa_set_int_o(P,insert_int(X,S4)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8764_finite__ranking__induct,axiom,
    ! [S: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_num] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X: vEBT_VEBT,S4: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
             => ( ! [Y4: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_VEBT_VEBT_num(F,Y4)),aa_VEBT_VEBT_num(F,X)) )
               => ( aa_set_VEBT_VEBT_o(P,S4)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X,S4)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8765_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_o,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X: complex,S4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S4)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,Y4)),aa_complex_num(F,X)) )
               => ( aa_set_complex_o(P,S4)
                 => aa_set_complex_o(P,insert_complex(X,S4)) ) ) )
         => aa_set_complex_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8766_finite__ranking__induct,axiom,
    ! [S: set_real,P: fun_set_real_o,F: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X: real,S4: set_real] :
              ( aa_set_real_o(finite_finite_real,S4)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,Y4)),aa_real_num(F,X)) )
               => ( aa_set_real_o(P,S4)
                 => aa_set_real_o(P,insert_real(X,S4)) ) ) )
         => aa_set_real_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8767_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X: nat,S4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S4)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Y4)),aa_nat_num(F,X)) )
               => ( aa_set_nat_o(P,S4)
                 => aa_set_nat_o(P,insert_nat(X,S4)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8768_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X: int,S4: set_int] :
              ( aa_set_int_o(finite_finite_int,S4)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,Y4)),aa_int_num(F,X)) )
               => ( aa_set_int_o(P,S4)
                 => aa_set_int_o(P,insert_int(X,S4)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8769_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B5: real,A6: set_real] :
              ( aa_set_real_o(finite_finite_real,A6)
             => ( ! [X5: real] :
                    ( aa_set_real_o(member_real(X5),A6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X5),B5) )
               => ( aa_set_real_o(P,A6)
                 => aa_set_real_o(P,insert_real(B5,A6)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8770_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B5: rat,A6: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A6)
             => ( ! [X5: rat] :
                    ( aa_set_rat_o(member_rat(X5),A6)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),B5) )
               => ( aa_set_rat_o(P,A6)
                 => aa_set_rat_o(P,insert_rat(B5,A6)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8771_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B5: num,A6: set_num] :
              ( aa_set_num_o(finite_finite_num,A6)
             => ( ! [X5: num] :
                    ( aa_set_num_o(member_num(X5),A6)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X5),B5) )
               => ( aa_set_num_o(P,A6)
                 => aa_set_num_o(P,insert_num(B5,A6)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8772_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B5: nat,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ! [X5: nat] :
                    ( aa_set_nat_o(member_nat(X5),A6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),B5) )
               => ( aa_set_nat_o(P,A6)
                 => aa_set_nat_o(P,insert_nat(B5,A6)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8773_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B5: int,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ! [X5: int] :
                    ( aa_set_int_o(member_int(X5),A6)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X5),B5) )
               => ( aa_set_int_o(P,A6)
                 => aa_set_int_o(P,insert_int(B5,A6)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8774_finite__linorder__max__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B5: code_integer,A6: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A6)
             => ( ! [X5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A6)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),B5) )
               => ( aa_set5643387332196575111eger_o(P,A6)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B5,A6)) ) ) )
         => aa_set5643387332196575111eger_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8775_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B5: real,A6: set_real] :
              ( aa_set_real_o(finite_finite_real,A6)
             => ( ! [X5: real] :
                    ( aa_set_real_o(member_real(X5),A6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B5),X5) )
               => ( aa_set_real_o(P,A6)
                 => aa_set_real_o(P,insert_real(B5,A6)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8776_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B5: rat,A6: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A6)
             => ( ! [X5: rat] :
                    ( aa_set_rat_o(member_rat(X5),A6)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B5),X5) )
               => ( aa_set_rat_o(P,A6)
                 => aa_set_rat_o(P,insert_rat(B5,A6)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8777_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B5: num,A6: set_num] :
              ( aa_set_num_o(finite_finite_num,A6)
             => ( ! [X5: num] :
                    ( aa_set_num_o(member_num(X5),A6)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B5),X5) )
               => ( aa_set_num_o(P,A6)
                 => aa_set_num_o(P,insert_num(B5,A6)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8778_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B5: nat,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ! [X5: nat] :
                    ( aa_set_nat_o(member_nat(X5),A6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B5),X5) )
               => ( aa_set_nat_o(P,A6)
                 => aa_set_nat_o(P,insert_nat(B5,A6)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8779_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B5: int,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ! [X5: int] :
                    ( aa_set_int_o(member_int(X5),A6)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B5),X5) )
               => ( aa_set_int_o(P,A6)
                 => aa_set_int_o(P,insert_int(B5,A6)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8780_finite__linorder__min__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B5: code_integer,A6: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A6)
             => ( ! [X5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A6)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B5),X5) )
               => ( aa_set5643387332196575111eger_o(P,A6)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B5,A6)) ) ) )
         => aa_set5643387332196575111eger_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8781_Min_OboundedI,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A4) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),lattic3629708407755379051n_real(A2)) ) ) ) ).

% Min.boundedI
tff(fact_8782_Min_OboundedI,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( ! [A4: rat] :
              ( aa_set_rat_o(member_rat(A4),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A4) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic8086005427650270231in_rat(A2)) ) ) ) ).

% Min.boundedI
tff(fact_8783_Min_OboundedI,axiom,
    ! [A2: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ( ! [A4: num] :
              ( aa_set_num_o(member_num(A4),A2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),A4) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),lattic5278467273892544601in_num(A2)) ) ) ) ).

% Min.boundedI
tff(fact_8784_Min_OboundedI,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A4) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic8718645017227715691in_int(A2)) ) ) ) ).

% Min.boundedI
tff(fact_8785_Min_OboundedI,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A4: nat] :
              ( aa_set_nat_o(member_nat(A4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A4) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),lattic8721135487736765967in_nat(A2)) ) ) ) ).

% Min.boundedI
tff(fact_8786_Min_OboundedE,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),lattic3629708407755379051n_real(A2))
         => ! [A7: real] :
              ( aa_set_real_o(member_real(A7),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A7) ) ) ) ) ).

% Min.boundedE
tff(fact_8787_Min_OboundedE,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic8086005427650270231in_rat(A2))
         => ! [A7: rat] :
              ( aa_set_rat_o(member_rat(A7),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A7) ) ) ) ) ).

% Min.boundedE
tff(fact_8788_Min_OboundedE,axiom,
    ! [A2: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),lattic5278467273892544601in_num(A2))
         => ! [A7: num] :
              ( aa_set_num_o(member_num(A7),A2)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),A7) ) ) ) ) ).

% Min.boundedE
tff(fact_8789_Min_OboundedE,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic8718645017227715691in_int(A2))
         => ! [A7: int] :
              ( aa_set_int_o(member_int(A7),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A7) ) ) ) ) ).

% Min.boundedE
tff(fact_8790_Min_OboundedE,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),lattic8721135487736765967in_nat(A2))
         => ! [A7: nat] :
              ( aa_set_nat_o(member_nat(A7),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A7) ) ) ) ) ).

% Min.boundedE
tff(fact_8791_eq__Min__iff,axiom,
    ! [A2: set_real,M: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ( M = lattic3629708407755379051n_real(A2) )
        <=> ( aa_set_real_o(member_real(M),A2)
            & ! [X4: real] :
                ( aa_set_real_o(member_real(X4),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M),X4) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_8792_eq__Min__iff,axiom,
    ! [A2: set_rat,M: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( ( M = lattic8086005427650270231in_rat(A2) )
        <=> ( aa_set_rat_o(member_rat(M),A2)
            & ! [X4: rat] :
                ( aa_set_rat_o(member_rat(X4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),X4) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_8793_eq__Min__iff,axiom,
    ! [A2: set_num,M: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ( ( M = lattic5278467273892544601in_num(A2) )
        <=> ( aa_set_num_o(member_num(M),A2)
            & ! [X4: num] :
                ( aa_set_num_o(member_num(X4),A2)
               => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),X4) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_8794_eq__Min__iff,axiom,
    ! [A2: set_int,M: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ( M = lattic8718645017227715691in_int(A2) )
        <=> ( aa_set_int_o(member_int(M),A2)
            & ! [X4: int] :
                ( aa_set_int_o(member_int(X4),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),X4) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_8795_eq__Min__iff,axiom,
    ! [A2: set_nat,M: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ( M = lattic8721135487736765967in_nat(A2) )
        <=> ( aa_set_nat_o(member_nat(M),A2)
            & ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),A2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),X4) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_8796_Min__le__iff,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic3629708407755379051n_real(A2)),Xa)
        <=> ? [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_8797_Min__le__iff,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(A2)),Xa)
        <=> ? [X4: rat] :
              ( aa_set_rat_o(member_rat(X4),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_8798_Min__le__iff,axiom,
    ! [A2: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic5278467273892544601in_num(A2)),Xa)
        <=> ? [X4: num] :
              ( aa_set_num_o(member_num(X4),A2)
              & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_8799_Min__le__iff,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(A2)),Xa)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_8800_Min__le__iff,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8721135487736765967in_nat(A2)),Xa)
        <=> ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_8801_Min__eq__iff,axiom,
    ! [A2: set_real,M: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ( lattic3629708407755379051n_real(A2) = M )
        <=> ( aa_set_real_o(member_real(M),A2)
            & ! [X4: real] :
                ( aa_set_real_o(member_real(X4),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M),X4) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_8802_Min__eq__iff,axiom,
    ! [A2: set_rat,M: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( ( lattic8086005427650270231in_rat(A2) = M )
        <=> ( aa_set_rat_o(member_rat(M),A2)
            & ! [X4: rat] :
                ( aa_set_rat_o(member_rat(X4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),X4) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_8803_Min__eq__iff,axiom,
    ! [A2: set_num,M: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ( ( lattic5278467273892544601in_num(A2) = M )
        <=> ( aa_set_num_o(member_num(M),A2)
            & ! [X4: num] :
                ( aa_set_num_o(member_num(X4),A2)
               => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),X4) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_8804_Min__eq__iff,axiom,
    ! [A2: set_int,M: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ( lattic8718645017227715691in_int(A2) = M )
        <=> ( aa_set_int_o(member_int(M),A2)
            & ! [X4: int] :
                ( aa_set_int_o(member_int(X4),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),X4) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_8805_Min__eq__iff,axiom,
    ! [A2: set_nat,M: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ( lattic8721135487736765967in_nat(A2) = M )
        <=> ( aa_set_nat_o(member_nat(M),A2)
            & ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),A2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),X4) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_8806_Min__less__iff,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,lattic3629708407755379051n_real(A2)),Xa)
        <=> ? [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_8807_Min__less__iff,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,lattic8086005427650270231in_rat(A2)),Xa)
        <=> ? [X4: rat] :
              ( aa_set_rat_o(member_rat(X4),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_8808_Min__less__iff,axiom,
    ! [A2: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,lattic5278467273892544601in_num(A2)),Xa)
        <=> ? [X4: num] :
              ( aa_set_num_o(member_num(X4),A2)
              & aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_8809_Min__less__iff,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,lattic8718645017227715691in_int(A2)),Xa)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_8810_Min__less__iff,axiom,
    ! [A2: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,lattic1063845414844153500nteger(A2)),Xa)
        <=> ? [X4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A2)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X4),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_8811_Min__less__iff,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,lattic8721135487736765967in_nat(A2)),Xa)
        <=> ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_8812_Min__insert2,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [B5: real] :
            ( aa_set_real_o(member_real(B5),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B5) )
       => ( lattic3629708407755379051n_real(insert_real(A,A2)) = A ) ) ) ).

% Min_insert2
tff(fact_8813_Min__insert2,axiom,
    ! [A2: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ! [B5: rat] :
            ( aa_set_rat_o(member_rat(B5),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B5) )
       => ( lattic8086005427650270231in_rat(insert_rat(A,A2)) = A ) ) ) ).

% Min_insert2
tff(fact_8814_Min__insert2,axiom,
    ! [A2: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ! [B5: num] :
            ( aa_set_num_o(member_num(B5),A2)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B5) )
       => ( lattic5278467273892544601in_num(insert_num(A,A2)) = A ) ) ) ).

% Min_insert2
tff(fact_8815_Min__insert2,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [B5: int] :
            ( aa_set_int_o(member_int(B5),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B5) )
       => ( lattic8718645017227715691in_int(insert_int(A,A2)) = A ) ) ) ).

% Min_insert2
tff(fact_8816_Min__insert2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [B5: nat] :
            ( aa_set_nat_o(member_nat(B5),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B5) )
       => ( lattic8721135487736765967in_nat(insert_nat(A,A2)) = A ) ) ) ).

% Min_insert2
tff(fact_8817_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ga,M))) ) ).

% finite_divisors_nat
tff(fact_8818_tan__def,axiom,
    ! [X5: complex] : tan_complex(X5) = divide1717551699836669952omplex(sin_complex(X5),cos_complex(X5)) ).

% tan_def
tff(fact_8819_tan__def,axiom,
    ! [X5: real] : aa_real_real(tan_real,X5) = aa_real_real(divide_divide_real(aa_real_real(sin_real,X5)),aa_real_real(cos_real,X5)) ).

% tan_def
tff(fact_8820_Min__antimono,axiom,
    ! [M7: set_real,N3: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(M7),N3)
     => ( ( M7 != bot_bot_set_real )
       => ( aa_set_real_o(finite_finite_real,N3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic3629708407755379051n_real(N3)),lattic3629708407755379051n_real(M7)) ) ) ) ).

% Min_antimono
tff(fact_8821_Min__antimono,axiom,
    ! [M7: set_rat,N3: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(M7),N3)
     => ( ( M7 != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,N3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(N3)),lattic8086005427650270231in_rat(M7)) ) ) ) ).

% Min_antimono
tff(fact_8822_Min__antimono,axiom,
    ! [M7: set_num,N3: set_num] :
      ( aa_set_num_o(ord_less_eq_set_num(M7),N3)
     => ( ( M7 != bot_bot_set_num )
       => ( aa_set_num_o(finite_finite_num,N3)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic5278467273892544601in_num(N3)),lattic5278467273892544601in_num(M7)) ) ) ) ).

% Min_antimono
tff(fact_8823_Min__antimono,axiom,
    ! [M7: set_int,N3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(M7),N3)
     => ( ( M7 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,N3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(N3)),lattic8718645017227715691in_int(M7)) ) ) ) ).

% Min_antimono
tff(fact_8824_Min__antimono,axiom,
    ! [M7: set_nat,N3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(M7),N3)
     => ( ( M7 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8721135487736765967in_nat(N3)),lattic8721135487736765967in_nat(M7)) ) ) ) ).

% Min_antimono
tff(fact_8825_Min_Osubset__imp,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_set_real_o(finite_finite_real,B2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic3629708407755379051n_real(B2)),lattic3629708407755379051n_real(A2)) ) ) ) ).

% Min.subset_imp
tff(fact_8826_Min_Osubset__imp,axiom,
    ! [A2: set_rat,B2: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(A2),B2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,B2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(B2)),lattic8086005427650270231in_rat(A2)) ) ) ) ).

% Min.subset_imp
tff(fact_8827_Min_Osubset__imp,axiom,
    ! [A2: set_num,B2: set_num] :
      ( aa_set_num_o(ord_less_eq_set_num(A2),B2)
     => ( ( A2 != bot_bot_set_num )
       => ( aa_set_num_o(finite_finite_num,B2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic5278467273892544601in_num(B2)),lattic5278467273892544601in_num(A2)) ) ) ) ).

% Min.subset_imp
tff(fact_8828_Min_Osubset__imp,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(B2)),lattic8718645017227715691in_int(A2)) ) ) ) ).

% Min.subset_imp
tff(fact_8829_Min_Osubset__imp,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8721135487736765967in_nat(B2)),lattic8721135487736765967in_nat(A2)) ) ) ) ).

% Min.subset_imp
tff(fact_8830_finite__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => aa_set_real_o(finite_finite_real,collect_real(aTP_Lamm_gy(Na))) ) ).

% finite_roots_unity
tff(fact_8831_finite__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_gz(Na))) ) ).

% finite_roots_unity
tff(fact_8832_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_8833_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),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,Y),aa_real_real(tan_real,X)) ) ) ).

% lemma_tan_total
tff(fact_8834_tan__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) ) ) ).

% tan_gt_zero
tff(fact_8835_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X: 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))))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X) = Y ) ) ).

% lemma_tan_total1
tff(fact_8836_tan__mono__lt__eq,axiom,
    ! [Xa: real,Y: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),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,Xa)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8837_tan__monotone_H,axiom,
    ! [Y: real,Xa: 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))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),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))))),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Y),Xa)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xa)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8838_tan__monotone,axiom,
    ! [Y: real,Xa: 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))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Y)),aa_real_real(tan_real,Xa)) ) ) ) ).

% tan_monotone
tff(fact_8839_tan__total,axiom,
    ! [Y: real] :
    ? [X: 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))))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X) = Y )
      & ! [Y4: 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))))),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X ) ) ) ).

% tan_total
tff(fact_8840_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_8841_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(divide_divide_real(one_one_real),aa_real_real(tan_real,Y)) = 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)))),Y)) ).

% tan_inverse
tff(fact_8842_add__tan__eq,axiom,
    ! [Xa: complex,Y: complex] :
      ( ( cos_complex(Xa) != zero_zero_complex )
     => ( ( cos_complex(Y) != zero_zero_complex )
       => ( plus_plus_complex(tan_complex(Xa),tan_complex(Y)) = divide1717551699836669952omplex(sin_complex(plus_plus_complex(Xa,Y)),aa_complex_complex(times_times_complex(cos_complex(Xa)),cos_complex(Y))) ) ) ) ).

% add_tan_eq
tff(fact_8843_add__tan__eq,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y)) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_8844_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X) = Y ) ) ) ).

% tan_total_pos
tff(fact_8845_tan__pos__pi2__le,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) ) ) ).

% tan_pos_pi2_le
tff(fact_8846_tan__less__zero,axiom,
    ! [Xa: 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)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xa)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_8847_tan__mono__le,axiom,
    ! [Xa: real,Y: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),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,Xa)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_8848_tan__mono__le__eq,axiom,
    ! [Xa: real,Y: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),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,Xa)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8849_tan__bound__pi2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),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,Xa))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_8850_arctan,axiom,
    ! [Y: 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,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),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,Y)) = Y ) ) ).

% arctan
tff(fact_8851_arctan__tan,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) = Xa ) ) ) ).

% arctan_tan
tff(fact_8852_arctan__unique,axiom,
    ! [Xa: real,Y: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xa) = Y )
         => ( aa_real_real(arctan,Y) = Xa ) ) ) ) ).

% arctan_unique
tff(fact_8853_tan__add,axiom,
    ! [Xa: complex,Y: complex] :
      ( ( cos_complex(Xa) != zero_zero_complex )
     => ( ( cos_complex(Y) != zero_zero_complex )
       => ( ( cos_complex(plus_plus_complex(Xa,Y)) != zero_zero_complex )
         => ( tan_complex(plus_plus_complex(Xa,Y)) = divide1717551699836669952omplex(plus_plus_complex(tan_complex(Xa),tan_complex(Y)),minus_minus_complex(one_one_complex,aa_complex_complex(times_times_complex(tan_complex(Xa)),tan_complex(Y)))) ) ) ) ) ).

% tan_add
tff(fact_8854_tan__add,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y))),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_8855_tan__diff,axiom,
    ! [Xa: complex,Y: complex] :
      ( ( cos_complex(Xa) != zero_zero_complex )
     => ( ( cos_complex(Y) != zero_zero_complex )
       => ( ( cos_complex(minus_minus_complex(Xa,Y)) != zero_zero_complex )
         => ( tan_complex(minus_minus_complex(Xa,Y)) = divide1717551699836669952omplex(minus_minus_complex(tan_complex(Xa),tan_complex(Y)),plus_plus_complex(one_one_complex,aa_complex_complex(times_times_complex(tan_complex(Xa)),tan_complex(Y)))) ) ) ) ) ).

% tan_diff
tff(fact_8856_tan__diff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(minus_minus_real(Xa),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(minus_minus_real(Xa),Y)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_8857_lemma__tan__add1,axiom,
    ! [Xa: complex,Y: complex] :
      ( ( cos_complex(Xa) != zero_zero_complex )
     => ( ( cos_complex(Y) != zero_zero_complex )
       => ( minus_minus_complex(one_one_complex,aa_complex_complex(times_times_complex(tan_complex(Xa)),tan_complex(Y))) = divide1717551699836669952omplex(cos_complex(plus_plus_complex(Xa,Y)),aa_complex_complex(times_times_complex(cos_complex(Xa)),cos_complex(Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_8858_lemma__tan__add1,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y))) = aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_8859_tan__total__pi4,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),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) = Xa ) ) ) ).

% tan_total_pi4
tff(fact_8860_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: fun_o_assn,R2: assn] :
      ( hoare_hoare_triple_o(P,C,Q)
     => hoare_hoare_triple_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R2),C,aa_assn_fun_o_assn(aTP_Lamm_ha(Q),R2)) ) ).

% frame_rule
tff(fact_8861_frame__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R2: assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R2),C,aa_ass4112278883933122603t_assn(aTP_Lamm_hb(Q),R2)) ) ).

% frame_rule
tff(fact_8862_frame__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R2: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R2),C,aa_ass4389913640333755968i_assn(aTP_Lamm_hc(Q),R2)) ) ).

% frame_rule
tff(fact_8863_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,R2: assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R2),C,aa_assn_fun_nat_assn(aTP_Lamm_hd(Q),R2)) ) ).

% frame_rule
tff(fact_8864_tan__half,axiom,
    ! [Xa: complex] : tan_complex(Xa) = divide1717551699836669952omplex(sin_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)),plus_plus_complex(cos_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)),one_one_complex)) ).

% tan_half
tff(fact_8865_tan__half,axiom,
    ! [Xa: real] : aa_real_real(tan_real,Xa) = 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))),Xa))),aa_real_real(aa_rea7340605660300375871l_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))),Xa))),one_one_real)) ).

% tan_half
tff(fact_8866_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,B2)))
    <=> aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) ) ).

% finite_Diff_insert
tff(fact_8867_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_he,K))) ).

% finite_Collect_le_nat
tff(fact_8868_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_hf,K))) ).

% finite_Collect_less_nat
tff(fact_8869_finite__interval__int1,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_hg(A),B))) ).

% finite_interval_int1
tff(fact_8870_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_hh(A),B))) ).

% finite_interval_int4
tff(fact_8871_finite__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_hi(Na),C))) ) ).

% finite_nth_roots
tff(fact_8872_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_hj(A),B))) ).

% finite_interval_int2
tff(fact_8873_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_hk(A),B))) ).

% finite_interval_int3
tff(fact_8874_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_fz,I))) ) ).

% finite_divisors_int
tff(fact_8875_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_8876_pure__false,axiom,
    aa_o_assn(pure_assn,$false) = bot_bot_assn ).

% pure_false
tff(fact_8877_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = bot_bot_assn )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_8878_star__false__left,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),P) = bot_bot_assn ).

% star_false_left
tff(fact_8879_star__false__right,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_8880_set__encode__insert,axiom,
    ! [A2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Na),A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Na,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_8881_sin__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xa) = aa_real_real(divide_divide_real(aa_real_real(tan_real,Xa)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_8882_real__sqrt__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(sqrt,Xa) = aa_real_real(sqrt,Y) )
    <=> ( Xa = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_8883_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_8884_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sqrt,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_8885_real__sqrt__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% real_sqrt_less_iff
tff(fact_8886_real__sqrt__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% real_sqrt_le_iff
tff(fact_8887_real__sqrt__eq__1__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sqrt,Xa) = one_one_real )
    <=> ( Xa = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_8888_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_8889_real__sqrt__lt__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_8890_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_8891_real__sqrt__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_8892_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_8893_real__sqrt__lt__1__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_8894_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_8895_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_8896_real__sqrt__le__1__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_8897_real__sqrt__abs2,axiom,
    ! [Xa: real] : aa_real_real(sqrt,aa_real_real(times_times_real(Xa),Xa)) = aa_real_real(abs_abs_real,Xa) ).

% real_sqrt_abs2
tff(fact_8898_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).

% real_sqrt_mult_self
tff(fact_8899_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_8900_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_8901_real__sqrt__abs,axiom,
    ! [Xa: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xa) ).

% real_sqrt_abs
tff(fact_8902_real__sqrt__pow2__iff,axiom,
    ! [Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% real_sqrt_pow2_iff
tff(fact_8903_real__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa ) ) ).

% real_sqrt_pow2
tff(fact_8904_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xa: real,Y: real,Xaa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_8905_real__sqrt__power,axiom,
    ! [Xa: real,K: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),K) ).

% real_sqrt_power
tff(fact_8906_real__sqrt__minus,axiom,
    ! [Xa: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,Xa)) ).

% real_sqrt_minus
tff(fact_8907_real__sqrt__mult,axiom,
    ! [Xa: real,Y: real] : aa_real_real(sqrt,aa_real_real(times_times_real(Xa),Y)) = aa_real_real(times_times_real(aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_8908_real__sqrt__divide,axiom,
    ! [Xa: real,Y: real] : aa_real_real(sqrt,aa_real_real(divide_divide_real(Xa),Y)) = aa_real_real(divide_divide_real(aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y)) ).

% real_sqrt_divide
tff(fact_8909_real__sqrt__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_8910_real__sqrt__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_8911_real__sqrt__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_gt_zero
tff(fact_8912_real__sqrt__eq__zero__cancel,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(sqrt,Xa) = zero_zero_real )
       => ( Xa = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_8913_real__sqrt__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_ge_zero
tff(fact_8914_real__sqrt__ge__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_ge_one
tff(fact_8915_real__div__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(divide_divide_real(Xa),aa_real_real(sqrt,Xa)) = aa_real_real(sqrt,Xa) ) ) ).

% real_div_sqrt
tff(fact_8916_sqrt__add__le__add__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_8917_le__real__sqrt__sumsq,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xa),Xa)),aa_real_real(times_times_real(Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_8918_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_8919_real__less__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_8920_real__le__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_8921_sqrt__le__D,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_8922_real__sqrt__unique,axiom,
    ! [Y: real,Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,Xa) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_8923_real__le__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_8924_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_8925_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( Xa = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_8926_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Xa )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_8927_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_8928_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_8929_real__sqrt__sum__squares__ge1,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_8930_sqrt__ge__absD,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_8931_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_8932_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_8933_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_8934_real__less__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_8935_sqrt__sum__squares__le__sum,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_8936_sqrt__even__pow2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( 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))),Na)) = 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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_8937_real__sqrt__ge__abs1,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_8938_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_8939_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_8940_ln__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xa)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Xa)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_8941_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_8942_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_8943_real__sqrt__power__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_8944_arsinh__real__aux,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_8945_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xa: real,Y: real,Xaa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_8946_arith__geo__mean__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(times_times_real(Xa),Y))),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_8947_powr__half__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xa) ) ) ).

% powr_half_sqrt
tff(fact_8948_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_8949_arsinh__real__def,axiom,
    ! [Xa: real] : aa_real_real(arsinh_real,Xa) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_8950_cos__x__y__le__one,axiom,
    ! [Xa: real,Y: 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(Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_8951_real__sqrt__sum__squares__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),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,Y)),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(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_8952_arcosh__real__def,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_real(arcosh_real,Xa) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_8953_cos__arctan,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(arctan,Xa)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_8954_sin__arctan,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(arctan,Xa)) = aa_real_real(divide_divide_real(Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_8955_sqrt__sum__squares__half__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Y),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),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_8956_sin__cos__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xa))
     => ( aa_real_real(sin_real,Xa) = 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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_8957_arctan__half,axiom,
    ! [Xa: real] : aa_real_real(arctan,Xa) = 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(Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_8958_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_8959_cos__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xa) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_8960_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ! [M3: int] :
        ? [N2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M3),aa_int_int(abs_abs_int,N2))
          & aa_set_int_o(member_int(N2),S) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8961_cos__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xa)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_8962_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_8963_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_8964_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_8965_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_8966_arcsin__le__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_8967_arcsin__minus,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xa)) ) ) ) ).

% arcsin_minus
tff(fact_8968_arcsin__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_8969_arcsin__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( ( aa_real_real(arcsin,Xa) = aa_real_real(arcsin,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_8970_arcsin__less__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_8971_arcsin__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_8972_cos__arcsin__nonzero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xa)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_8973_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M4)
         => ? [N9: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N9)
              & aa_set_nat_o(member_nat(N9),S) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S) ) ).

% unbounded_k_infinite
tff(fact_8974_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
    <=> ! [M3: nat] :
        ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
          & aa_set_nat_o(member_nat(N2),S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_8975_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_8976_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_8977_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_8978_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_8979_arcsin__sin,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)) = Xa ) ) ) ).

% arcsin_sin
tff(fact_8980_le__arcsin__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),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,Y),aa_real_real(arcsin,Xa))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xa) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_8981_arcsin__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),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,Xa)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_8982_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_8983_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_8984_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_8985_sin__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xa)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_8986_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_8987_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_8988_cos__arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_8989_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_8990_arccos__le__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xa)) ) ) ) ).

% arccos_le_arccos
tff(fact_8991_arccos__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real) )
     => ( ( aa_real_real(arccos,Xa) = aa_real_real(arccos,Y) )
      <=> ( Xa = Y ) ) ) ).

% arccos_eq_iff
tff(fact_8992_arccos__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xa)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ) ).

% arccos_le_mono
tff(fact_8993_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_8994_arccos__less__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xa)) ) ) ) ).

% arccos_less_arccos
tff(fact_8995_arccos__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xa)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ) ).

% arccos_less_mono
tff(fact_8996_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_8997_arccos__cos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xa)) = Xa ) ) ) ).

% arccos_cos
tff(fact_8998_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_8999_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_9000_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_9001_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_9002_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_9003_sin__arccos__nonzero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xa)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9004_arccos__cos2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xa)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xa)) = aa_real_real(uminus_uminus_real,Xa) ) ) ) ).

% arccos_cos2
tff(fact_9005_arccos__minus,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,Xa)) ) ) ) ).

% arccos_minus
tff(fact_9006_arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_9007_arccos__minus__abs,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,Xa)) ) ) ).

% arccos_minus_abs
tff(fact_9008_mask__eq__sum__exp__nat,axiom,
    ! [Na: 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))),Na)),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_hf,Na))) ).

% mask_eq_sum_exp_nat
tff(fact_9009_gauss__sum__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cg),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_9010_arith__series__nat,axiom,
    ! [A: nat,D: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hl(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = 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,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_9011_Sum__Icc__nat,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cg),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_9012_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_9013_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: 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,K2)),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_9014_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_9015_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Na) = bot_bot_set_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_9016_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_9017_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_9018_sum__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_hm,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_hi(Na),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_9019_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_hm,collect_complex(aTP_Lamm_gz(Na))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_9020_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_hn),one_one_real) ).

% power_half_series
tff(fact_9021_sums__if_H,axiom,
    ! [G: fun_nat_real,Xa: real] :
      ( aa_real_o(sums_real(G),Xa)
     => aa_real_o(sums_real(aTP_Lamm_ho(G)),Xa) ) ).

% sums_if'
tff(fact_9022_sums__if,axiom,
    ! [G: fun_nat_real,Xa: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),Xa)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_hp(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) ) ) ).

% sums_if
tff(fact_9023_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Na: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_hq(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))),Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_hr(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))),groups6591440286371151544t_real(aTP_Lamm_hs(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum_split_even_odd
tff(fact_9024_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),Na)
     => ( groups4538972089207619220nt_int(aTP_Lamm_ch,set_or1266510415728281911st_int(M,Na)) = 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,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_9025_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_9026_sumr__cos__zero__one,axiom,
    ! [Na: nat] : groups6591440286371151544t_real(aTP_Lamm_ht,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_9027_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_9028_Maclaurin__cos__expansion2,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xa)
            & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hu,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_9029_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
            & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hu,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_9030_Maclaurin__cos__expansion,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hu,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9031_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = groups1705073143266064639nt_int(aTP_Lamm_ch,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_9032_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_ch,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_9033_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,Na))),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))),Na))) ).

% square_fact_le_2_fact
tff(fact_9034_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B3: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_hv(H),J),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(B3),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na)),aa_nat_real(semiri2265585572941072030t_real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_9035_cos__coeff__def,axiom,
    ! [X5: nat] :
      aa_nat_real(cos_coeff,X5) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5),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,X5),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X5)),zero_zero_real) ).

% cos_coeff_def
tff(fact_9036_cos__paired,axiom,
    ! [Xa: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_hw,Xa)),aa_real_real(cos_real,Xa)) ).

% cos_paired
tff(fact_9037_sin__paired,axiom,
    ! [Xa: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_hx,Xa)),aa_real_real(sin_real,Xa)) ).

% sin_paired
tff(fact_9038_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xa)
            & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hy,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_9039_Maclaurin__sin__expansion4,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),Xa)
          & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hy,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_9040_Maclaurin__sin__expansion2,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hy,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9041_Maclaurin__sin__expansion,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] : aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hy,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ).

% Maclaurin_sin_expansion
tff(fact_9042_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_9043_fact__less__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_9044_diffs__sin__coeff,axiom,
    diffs_real(sin_coeff) = cos_coeff ).

% diffs_sin_coeff
tff(fact_9045_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_9046_fact__div__fact__le__pow,axiom,
    ! [R: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),Na)
     => 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,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),R)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R)) ) ).

% fact_div_fact_le_pow
tff(fact_9047_diffs__cos__coeff,axiom,
    ! [X5: nat] : aa_nat_real(diffs_real(cos_coeff),X5) = aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,X5)) ).

% diffs_cos_coeff
tff(fact_9048_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),groups708209901874060359at_nat(aTP_Lamm_cg,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Na)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_9049_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = groups708209901874060359at_nat(aTP_Lamm_cg,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_9050_sin__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Na)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% sin_coeff_Suc
tff(fact_9051_cos__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Na)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Na))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% cos_coeff_Suc
tff(fact_9052_sin__coeff__def,axiom,
    ! [X5: nat] :
      aa_nat_real(sin_coeff,X5) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5),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,X5),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X5))) ).

% sin_coeff_def
tff(fact_9053_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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,Na),K)),one_one_nat)),Na),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_9054_Maclaurin__exp__lt,axiom,
    ! [Xa: real,Na: nat] :
      ( ( Xa != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(exp_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hz,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_9055_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Na: 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,Na) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),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,Na),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_9056_maxt__sound,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),Xa)
       => ( vEBT_vebt_maxt(T) = some_nat(Xa) ) ) ) ).

% maxt_sound
tff(fact_9057_maxbmo,axiom,
    ! [T: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_maxt(T) = some_nat(Xa) )
     => aa_nat_o(vEBT_V8194947554948674370ptions(T),Xa) ) ).

% maxbmo
tff(fact_9058_maxt__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(T),Maxi) ) ) ).

% maxt_member
tff(fact_9059_exp__inj__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(exp_real,Xa) = aa_real_real(exp_real,Y) )
    <=> ( Xa = Y ) ) ).

% exp_inj_iff
tff(fact_9060_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( aa_nat_o(vEBT_vebt_member(T),Xa)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_9061_set__vebt__maxt_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Xa) )
      <=> ( aa_set_nat_o(member_nat(Xa),vEBT_set_vebt(T))
          & ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),vEBT_set_vebt(T))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa) ) ) ) ) ).

% set_vebt_maxt'
tff(fact_9062_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_9063_set__vebt__maxt,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Xa) )
      <=> aa_nat_o(vEBT_VEBT_max_in_set(vEBT_set_vebt(T)),Xa) ) ) ).

% set_vebt_maxt
tff(fact_9064_maxt__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Xa) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),Xa) ) ) ).

% maxt_corr
tff(fact_9065_exp__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% exp_less_cancel_iff
tff(fact_9066_exp__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y)) ) ).

% exp_less_mono
tff(fact_9067_exp__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% exp_le_cancel_iff
tff(fact_9068_abs__exp__cancel,axiom,
    ! [Xa: real] : aa_real_real(abs_abs_real,aa_real_real(exp_real,Xa)) = aa_real_real(exp_real,Xa) ).

% abs_exp_cancel
tff(fact_9069_ln__exp,axiom,
    ! [Xa: real] : aa_real_real(ln_ln_real,aa_real_real(exp_real,Xa)) = Xa ).

% ln_exp
tff(fact_9070_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% binomial_eq_0_iff
tff(fact_9071_binomial__Suc__Suc,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),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,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_9072_exp__eq__one__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(exp_real,Xa) = one_one_real )
    <=> ( Xa = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_9073_zero__less__binomial__iff,axiom,
    ! [Na: 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,Na),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% zero_less_binomial_iff
tff(fact_9074_one__less__exp__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% one_less_exp_iff
tff(fact_9075_exp__less__one__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_9076_exp__le__one__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_9077_one__le__exp__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% one_le_exp_iff
tff(fact_9078_exp__ln,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xa)) = Xa ) ) ).

% exp_ln
tff(fact_9079_exp__ln__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xa)) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% exp_ln_iff
tff(fact_9080_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_9081_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_9082_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),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,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_9083_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_9084_ln__unique,axiom,
    ! [Y: real,Xa: real] :
      ( ( aa_real_real(exp_real,Y) = Xa )
     => ( aa_real_real(ln_ln_real,Xa) = Y ) ) ).

% ln_unique
tff(fact_9085_exp__less__cancel,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% exp_less_cancel
tff(fact_9086_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_9087_Suc__times__binomial,axiom,
    ! [K: nat,Na: 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,Na)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ).

% Suc_times_binomial
tff(fact_9088_Suc__times__binomial__eq,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),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,Na)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_9089_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),M)) ).

% choose_mult_lemma
tff(fact_9090_binomial__le__pow,axiom,
    ! [R: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),R)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R)) ) ).

% binomial_le_pow
tff(fact_9091_exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X: real] : aa_real_real(exp_real,X) = Y ) ).

% exp_total
tff(fact_9092_exp__gt__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xa)) ).

% exp_gt_zero
tff(fact_9093_not__exp__less__zero,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_9094_exp__ge__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xa)) ).

% exp_ge_zero
tff(fact_9095_not__exp__le__zero,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_9096_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ).

% zero_less_binomial
tff(fact_9097_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_9098_binomial__Suc__Suc__eq__times,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),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,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_9099_choose__mult,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_9100_binomial__absorb__comp,axiom,
    ! [Na: 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,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_9101_exp__gt__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xa)) ) ).

% exp_gt_one
tff(fact_9102_exp__ge__add__one__self,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(exp_real,Xa)) ).

% exp_ge_add_one_self
tff(fact_9103_log__ln,axiom,
    ! [Xa: real] : aa_real_real(ln_ln_real,Xa) = aa_real_real(log(aa_real_real(exp_real,one_one_real)),Xa) ).

% log_ln
tff(fact_9104_binomial__absorption,axiom,
    ! [K: nat,Na: 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,Na),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_9105_nat__floor__neg,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9106_floor__eq3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = Na ) ) ) ).

% floor_eq3
tff(fact_9107_exp__ge__add__one__self__aux,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(exp_real,Xa)) ) ).

% exp_ge_add_one_self_aux
tff(fact_9108_le__nat__floor,axiom,
    ! [Xa: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_9109_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(minus_minus_real(Y),one_one_real))
          & ( aa_real_real(exp_real,X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_9110_ln__ge__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xa) ) ) ).

% ln_ge_iff
tff(fact_9111_ln__x__over__x__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Xa)),Xa)) ) ) ).

% ln_x_over_x_mono
tff(fact_9112_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( 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,Na),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_9113_floor__eq,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xa) = Na ) ) ) ).

% floor_eq
tff(fact_9114_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_9115_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_9116_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))) ).

% real_of_int_floor_gt_diff_one
tff(fact_9117_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))) ).

% real_of_int_floor_ge_diff_one
tff(fact_9118_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_9119_binomial__antimono,axiom,
    ! [K: nat,K4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_9120_binomial__mono,axiom,
    ! [K: nat,K4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K4)
     => ( 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))),K4)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K4)) ) ) ).

% binomial_mono
tff(fact_9121_binomial__maximum_H,axiom,
    ! [Na: 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))),Na)),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))),Na)),Na)) ).

% binomial_maximum'
tff(fact_9122_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% binomial_le_pow2
tff(fact_9123_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),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,Na),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,Na),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_9124_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: 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,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_9125_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_9126_floor__eq4,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = Na ) ) ) ).

% floor_eq4
tff(fact_9127_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),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,Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_9128_floor__eq2,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xa) = Na ) ) ) ).

% floor_eq2
tff(fact_9129_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_9130_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),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,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_9131_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K4)
     => ( 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))),K4)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K4)) ) ) ).

% binomial_strict_mono
tff(fact_9132_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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,K4),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_9133_central__binomial__odd,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_9134_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),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,Na),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,Na),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_9135_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_9136_choose__two,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),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,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_9137_exp__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_9138_real__exp__bound__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa))) ) ) ).

% real_exp_bound_lemma
tff(fact_9139_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(Xa),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,Xa)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_9140_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => 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(Xa),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xa))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_9141_floor__log__eq__powr__iff,axiom,
    ! [Xa: real,B: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( 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),Xa)) = 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))),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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_9142_Maclaurin__exp__le,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(exp_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hz,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_9143_exp__lower__Taylor__quadratic,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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,Xa)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_9144_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( 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,Na))) = 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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9145_log__base__10__eq2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xa) = 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,Xa)) ) ) ).

% log_base_10_eq2
tff(fact_9146_tanh__real__altdef,axiom,
    ! [Xa: real] : aa_real_real(tanh_real,Xa) = 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)))),Xa)))),aa_real_real(aa_rea7340605660300375871l_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)))),Xa)))) ).

% tanh_real_altdef
tff(fact_9147_floor__log__nat__eq__if,axiom,
    ! [B: nat,Na: 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),Na)),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,Na),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,Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9148_log__base__10__eq1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xa) = 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,Xa)) ) ) ).

% log_base_10_eq1
tff(fact_9149_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => 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)))),Na)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na)))),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))),Na)),Na))) ) ).

% central_binomial_lower_bound
tff(fact_9150_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_maxti(Tia),aa_VEB7611521244314475576t_assn(aTP_Lamm_ia(T),Tia)),one_one_nat) ).

% vebt_maxti_hT
tff(fact_9151_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_maxti(Tia),aa_VEB7611521244314475576t_assn(aTP_Lamm_ia(T),Tia)) ).

% vebt_maxti_h
tff(fact_9152_set__vebt__maxt_H_H,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( vEBT_vebt_maxt(T) = $ite(vEBT_set_vebt(T) = bot_bot_set_nat,none_nat,some_nat(lattic8265883725875713057ax_nat(vEBT_set_vebt(T)))) ) ) ).

% set_vebt_maxt''
tff(fact_9153_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_vebt_maxti(T)),one_one_nat) ).

% TBOUND_vebt_maxti
tff(fact_9154_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ( lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ga,Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_9155_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_9156_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_9157_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_9158_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ib,M)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_9159_divide__nat__def,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = $ite(Na = zero_zero_nat,zero_zero_nat,lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ic(M),Na)))) ).

% divide_nat_def
tff(fact_9160_sum__choose__lower,axiom,
    ! [R: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_id,R)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R),Na))),Na) ).

% sum_choose_lower
tff(fact_9161_choose__rising__sum_I1_J,axiom,
    ! [Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ie,Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9162_choose__rising__sum_I2_J,axiom,
    ! [Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ie,Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_9163_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_if(M),Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),M) ) ) ).

% sum_choose_diagonal
tff(fact_9164_vandermonde,axiom,
    ! [M: nat,Na: nat,R: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ig(M),Na),R)),aa_nat_set_nat(set_ord_atMost_nat,R)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),R) ).

% vandermonde
tff(fact_9165_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,Na)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9166_choose__row__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% choose_row_sum
tff(fact_9167_binomial,axiom,
    ! [A: nat,B: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Na) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ih(A),B),Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% binomial
tff(fact_9168_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,Na: nat,B: fun_nat_nat,Xa: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I3)
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J3)
           => ( aa_nat_nat(B,J3) = 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_ii(A),Xa)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ii(B),Xa)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ik(A),B),Xa)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_9169_choose__square__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_il,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = 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))),Na)),Na) ).

% choose_square_sum
tff(fact_9170_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_9171_choose__linear__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_im,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ).

% choose_linear_sum
tff(fact_9172_Max__divisors__self__int,axiom,
    ! [Na: int] :
      ( ( Na != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_fz,Na))) = aa_int_int(abs_abs_int,Na) ) ) ).

% Max_divisors_self_int
tff(fact_9173_real__scaleR__def,axiom,
    ! [A: real,Xa: real] : aa_real_real(real_V1485227260804924795R_real(A),Xa) = aa_real_real(times_times_real(A),Xa) ).

% real_scaleR_def
tff(fact_9174_height__compose__summary,axiom,
    ! [Summary2: vEBT_VEBT,Info2: option4927543243414619207at_nat,Deg: nat,TreeList: 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,Summary2))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Info2,Deg,TreeList,Summary2))) ).

% height_compose_summary
tff(fact_9175_Maclaurin__sin__bound,axiom,
    ! [Xa: real,Na: 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,Xa)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hy,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))))),aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xa)),Na))) ).

% Maclaurin_sin_bound
tff(fact_9176_deg__deg__n,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList,Summary2)),Na)
     => ( Deg = Na ) ) ).

% deg_deg_n
tff(fact_9177_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
     => ? [Info: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = vEBT_Node(Info,aa_nat_nat(suc,aa_nat_nat(suc,Na)),TreeList2,S2) ) ).

% deg_SUcn_Node
tff(fact_9178_real__sqrt__inverse,axiom,
    ! [Xa: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa)) ).

% real_sqrt_inverse
tff(fact_9179_divide__real__def,axiom,
    ! [Xa: real,Y: real] : aa_real_real(divide_divide_real(Xa),Y) = aa_real_real(times_times_real(Xa),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_9180_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_real(powr_real(aa_real_real(inverse_inverse_real,Y)),A) = aa_real_real(inverse_inverse_real,aa_real_real(powr_real(Y),A)) ) ) ).

% inverse_powr
tff(fact_9181_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [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_9182_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
    <=> ? [N2: nat] :
          ( ( N2 != 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,N2)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))),E) ) ) ).

% real_arch_inverse
tff(fact_9183_forall__pos__mono,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [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_9184_sqrt__divide__self__eq,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(divide_divide_real(aa_real_real(sqrt,Xa)),Xa) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9185_ln__inverse,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_inverse
tff(fact_9186_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,Xa: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),Xa) = 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_9187_log__inverse,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(log(A),aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(log(A),Xa)) ) ) ) ) ).

% log_inverse
tff(fact_9188_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,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: 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)),TreeList,Summary2)),Xa) = 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_9189_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,Xa: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xa) = 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_9190_exp__plus__inverse__exp,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))) ).

% exp_plus_inverse_exp
tff(fact_9191_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,Xa: 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)),Xa) = 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_9192_plus__inverse__ge__2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(inverse_inverse_real,Xa))) ) ).

% plus_inverse_ge_2
tff(fact_9193_real__inv__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xa) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9194_tan__cot,axiom,
    ! [Xa: 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)))),Xa)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xa)) ).

% tan_cot
tff(fact_9195_real__le__x__sinh,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_9196_real__le__abs__sinh,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_9197_powr__real__of__int,axiom,
    ! [Xa: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_int_real(ring_1_of_int_real,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,Na)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Na))))) ) ) ).

% powr_real_of_int
tff(fact_9198_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_VEBT_insert(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa) = $ite(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))),Deg)),Xa),vEBT_Node(Info2,Deg,TreeList,Summary2),vEBT_vebt_insert(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa)) ).

% VEBT_internal.insert'.simps(2)
tff(fact_9199_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Na)
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Ma)
          & ? [M4: nat] :
              ( ( some_nat(M4) = vEBT_vebt_mint(Summary2) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_9200_both__member__options__ding,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList,Summary2)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(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_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info2,Deg,TreeList,Summary2)),Xa) ) ) ) ).

% both_member_options_ding
tff(fact_9201_pred__max,axiom,
    ! [Deg: nat,Ma: nat,Xa: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = some_nat(Ma) ) ) ) ).

% pred_max
tff(fact_9202_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_9203_insert__simp__mima,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        | ( Xa = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2) ) ) ) ).

% insert_simp_mima
tff(fact_9204_delt__out__of__range,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2) ) ) ) ).

% delt_out_of_range
tff(fact_9205_del__single__cont,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & ( Xa = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2) ) ) ) ).

% del_single_cont
tff(fact_9206_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList,Summary2)),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% tdeletemimi
tff(fact_9207_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList,Summary2)),Xa)),one_one_nat) ) ).

% tdeletemimi'
tff(fact_9208_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Na)
     => ( 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_2_deg
tff(fact_9209_succ__min,axiom,
    ! [Deg: nat,Xa: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = some_nat(Mi) ) ) ) ).

% succ_min
tff(fact_9210_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Xa)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(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)))))
          | ( Xa = Mi )
          | ( Xa = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_9211_in__children__def,axiom,
    ! [Na: nat,TreeList: list_VEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Na,TreeList),Xa)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Na)) ) ).

% in_children_def
tff(fact_9212_div__half__nat,axiom,
    ! [Y: nat,Xa: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y)) = $let(
            q: nat,
            q:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)),
            $let(
              r: nat,
              r:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,q),Y)),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),r),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,q),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,r),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_9213_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( 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_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(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_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Xa) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_9214_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_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_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_9215_pred__list__to__short,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_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_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_9216_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
        & ( ( Xa = Mi )
          | ( Xa = Ma )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
            & 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_o(vEBT_vebt_member(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))))) ) ) ) ) ).

% member_inv
tff(fact_9217_xor__num_Ocases,axiom,
    ! [Xa: product_prod_num_num] :
      ( ( Xa != product_Pair_num_num(one,one) )
     => ( ! [N: num] : Xa != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : Xa != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M4: num] : Xa != product_Pair_num_num(aa_num_num(bit0,M4),one)
           => ( ! [M4: num,N: num] : Xa != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N))
             => ( ! [M4: num,N: num] : Xa != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N))
               => ( ! [M4: num] : Xa != product_Pair_num_num(aa_num_num(bit1,M4),one)
                 => ( ! [M4: num,N: num] : Xa != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N))
                   => ~ ! [M4: num,N: num] : Xa != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_9218_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,Sa: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,Sa)),Xa)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_9219_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd)),Xa)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_9220_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc)),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_9221_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xaa = Mi2 )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2)
             => ~ ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 )
                  | $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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_9222_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,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
        $ite(
          Xa = Mi,
          one_one_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
            $ite(
              Xa = 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,Xa),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),Xa),
                      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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_9223_vebt__member__code_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Xa)
    <=> $ite(
          ( ( Deg = zero_zero_nat )
          | ( Deg = aa_nat_nat(suc,zero_zero_nat) ) ),
          $false,
          $ite(
            Xa = Mi,
            $true,
            $ite(
              Xa = Ma,
              $true,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                $false,
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
                  $false,
                  $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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% vebt_member_code(3)
tff(fact_9224_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa)
    <=> $ite(
          Xa = Mi,
          $true,
          $ite(
            Xa = Ma,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_9225_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Na: nat,Va: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Na)
     => ( ( Na = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Mi)
         => ( ( Ma != Mi )
           => 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(Summary2))),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(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),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(TreeList)) ) ) ) ) ).

% nested_mint
tff(fact_9226_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Deg)
     => ( ( Mi != Ma )
       => ( the_nat(vEBT_vebt_maxt(Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_9227_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_9228_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_9229_num_Osize_I6_J,axiom,
    ! [X32: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_9230_del__x__mia,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $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(Summary2))),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(Summary2)))))),
                $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)),
                    $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,Deg),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(Summary2,h),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Ma ))),Deg,newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                $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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),Deg,newlist,Summary2) ) ) ),
                    vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_9231_del__x__mi__lets__in__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),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(Summary2)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summary2,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_9232_del__x__mi__lets__in,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),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(Summary2)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,H),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg,Newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_9233_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                   => ( 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))
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                $ite(Xa = 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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_9234_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),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(Summary2)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_9235_del__x__not__mia,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))
               => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $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),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,H),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Ma ))),Deg,newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                $ite(Xa = 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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Ma))),Deg,newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_9236_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary2: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summary2,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( 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))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none_nat,Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                    Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_9237_del__x__not__mi,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                 => ( 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))
                   => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,H),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                    Ma ))),Deg,Newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                $ite(Xa = 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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_9238_del__in__range,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( 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) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $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(Summary2))),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(Summary2)))))),Xa),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xa = 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,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)),
                      $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,Deg),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(Summary2,h),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg,newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),Deg,newlist,Summary2) ) ) ),
                      vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_9239_del__x__mi,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),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(Summary2)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $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),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,H),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg,newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Ma))),Deg,newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_9240_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) ),
        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),
        $ite(
          ( ( Xa = Mi )
          & ( Xa = Ma ) ),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),
          $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(Summary2))),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(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),Xa),
            $let(
              minn: nat,
              minn:= 
                $ite(Xa = 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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    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,Va))),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(Summary2,h),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_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,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)))))) ),
                                  Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist,sn) ),
                        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_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,Va))),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,Va)),newlist,Summary2) ) ) ),
                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_9241_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_V1232361888498592333_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) ),
        one_one_nat,
        $ite(
          ( ( Xa = Mi )
          & ( Xa = Ma ) ),
          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(Summary2))),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(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),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,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(TreeList)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(nth_VEBT_VEBT(TreeList,h)),l)),
                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),l)),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary2),h),one_one_nat)),
                  one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_9242_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,Xa: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),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_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),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,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_9243_max__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)) ).

% max_Suc_Suc
tff(fact_9244_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_9245_max__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A) = A ).

% max_nat.left_neutral
tff(fact_9246_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_9247_max__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),zero_zero_nat) = A ).

% max_nat.right_neutral
tff(fact_9248_max__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Na) = Na ).

% max_0L
tff(fact_9249_max__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),zero_zero_nat) = Na ).

% max_0R
tff(fact_9250_max__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_9251_max__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% max_numeral_Suc
tff(fact_9252_insert__simp__norm,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( 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_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Ma))),Deg,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_insert(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,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)))))),vEBT_vebt_insert(Summary2,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))))),Summary2)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_9253_nat__add__max__left,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Q2)) ).

% nat_add_max_left
tff(fact_9254_nat__add__max__right,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ).

% nat_add_max_right
tff(fact_9255_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9256_nat__mult__max__right,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_max_right
tff(fact_9257_nat__mult__max__left,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q2)) ).

% nat_mult_max_left
tff(fact_9258_nat__minus__add__max,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),M) ).

% nat_minus_add_max
tff(fact_9259_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Mi,Xa),
        $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)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Xa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),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,Va))),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(Summary2,h),Summary2)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_9260_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_9261_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_9262_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9263_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,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),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(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,Xa),Mi),Mi,Xa),
          $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(TreeList))
              & ~ ( ( Xa = Mi )
                  | ( Xa = 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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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(Summary2),h),one_one_nat)),
              one_one_nat ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_9264_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,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          Xa = Mi,
          zero_zero_nat,
          $ite(
            Xa = Ma,
            zero_zero_nat,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
              zero_zero_nat,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
                zero_zero_nat,
                $ite(
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) ),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                  zero_zero_nat ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_9265_vebt__maxtilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts))
     => hoare_7629718768684598413on_nat(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts,Tsi),vEBT_vebt_maxti(nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_in(I),Ts),Tsi)) ) ).

% vebt_maxtilist
tff(fact_9266_vebt__mintilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts))
     => hoare_7629718768684598413on_nat(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts,Tsi),vEBT_vebt_minti(nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_io(I),Ts),Tsi)) ) ).

% vebt_mintilist
tff(fact_9267_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_9268_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r2(T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% member_bound_height'
tff(fact_9269_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Xa),Xaa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
             => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2,Summary)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2,Summary)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                              one_one_nat,
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                one_one_nat,
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,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(TreeList2,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(TreeList2)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(nth_VEBT_VEBT(TreeList2,h)),l)),
                                          $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l)),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary),h),one_one_nat)),
                                        one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_9270_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
        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,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & 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(Summary2),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_9271_vebt__delete_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != vEBT_Leaf($false,(B5)) ) ) )
       => ( ! [A4: $o] :
              ( ? [B5: $o] : Xa = vEBT_Leaf((A4),(B5))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf((A4),$false) ) ) )
         => ( ! [A4: $o,B5: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B5)) )
               => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
                 => ( Y != vEBT_Leaf((A4),(B5)) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary) )
                 => ( Y != vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry) )
                   => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) )
                     => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,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(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi2,xn,Mi2),
                                    $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(TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,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(TreeList2,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(product_Pair_nat_nat(minn),
                                                      $ite(
                                                        xn = Ma2,
                                                        $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)))))) ),
                                                        Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,sn) ),
                                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                    $ite(xn = Ma2,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)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,Summary) ) ) ),
                                        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_9272_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_9273_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(T),one_one_nat)
    <=> ? [A8: $o,B8: $o] : T = vEBT_Leaf((A8),(B8)) ) ).

% deg1Leaf
tff(fact_9274_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(T),one_one_nat)
     => ? [A4: $o,B5: $o] : T = vEBT_Leaf((A4),(B5)) ) ).

% deg_1_Leaf
tff(fact_9275_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( Na = one_one_nat )
       => ? [A4: $o,B5: $o] : T = vEBT_Leaf((A4),(B5)) ) ) ).

% deg_1_Leafy
tff(fact_9276_greater__shift,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
    <=> vEBT_VEBT_greater(some_nat(Xa),some_nat(Y)) ) ).

% greater_shift
tff(fact_9277_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,vEBT_Leaf((A),(B))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.space'.simps(1)
tff(fact_9278_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((A),(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_9279_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_9280_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A),(B))),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(Xa = 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_9281_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,vEBT_Leaf((A),(B))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.space.simps(1)
tff(fact_9282_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_9283_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A),(B))),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
        $ite(Xa = 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_9284_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(Xa,Xaa) = Y )
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => ( Y != vEBT_vebt_insert(vEBT_Leaf((A4),(B5)),Xaa) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
             => ( Y != $ite(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))),Deg2)),Xaa),vEBT_Node(Info,Deg2,TreeList2,Summary),vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList2,Summary),Xaa)) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_9285_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xa),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,Uw2: $o] : Xa = vEBT_Leaf((A4),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
             => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V3: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & 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(TreeList2,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_9286_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B5),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S2)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_9287_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B5),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_9288_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B5),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_9289_vebt__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B5),$false) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary)
             => ~ $ite(
                    Xaa = Mi2,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_9290_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xa != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2)
                 => ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 )
                    | $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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_9291_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2)
           => (Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_9292_vebt__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B5),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
           => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary)
                   => $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_9293_vebt__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B5),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
             => (Y) )
           => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_9294_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xa),Xaa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != one_one_nat ) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                          $ite(
                            Xaa = Mi2,
                            zero_zero_nat,
                            $ite(
                              Xaa = Ma2,
                              zero_zero_nat,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                zero_zero_nat,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  zero_zero_nat,
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa)
                                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma2) ),
                                    $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(TreeList2)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList2,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_9295_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xa),Xaa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
             => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
           => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $ite(
                            Xaa = Mi2,
                            one_one_nat,
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Ma2,
                                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),Mi2),
                                    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,Ma2),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(TreeList2)),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(TreeList2,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_9296_vebt__insert_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B5)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B5))) ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,zero_zero_nat,Ts2,S2) )
             => ( Y != vEBT_Node(Info,zero_zero_nat,Ts2,S2) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
               => ( Y != vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) ) )
           => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,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)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Xaa,Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,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(TreeList2,h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_9297_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xa),Xaa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xa = vEBT_Node(Info,zero_zero_nat,Ts2,S2)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xa = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary)
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,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(TreeList2))
                                & ~ ( ( Xaa = Mi2 )
                                    | ( Xaa = Ma2 ) ) ),
                                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(TreeList2,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)))))),vEBT_T_m_i_n_N_u_l_l(nth_VEBT_VEBT(TreeList2,h)))),
                                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,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_9298_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xa),Xaa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( Y != one_one_nat ) )
       => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xa = vEBT_Node(Info,zero_zero_nat,Ts2,S2)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xa = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,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(TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              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(TreeList2,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(TreeList2,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_9299_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xa),Xaa) = Y )
     => ( ( ? [Uu2: $o,B5: $o] : Xa = vEBT_Leaf((Uu2),(B5))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xa = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            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(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & 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(TreeList2,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_9300_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
        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,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & 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(Summary2),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_9301_less__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> vEBT_VEBT_less(some_nat(Xa),some_nat(Y)) ) ).

% less_shift
tff(fact_9302_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% insert'_bound_height
tff(fact_9303_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,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Mi,Xa),
        $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(TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = 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,Va))),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(Summary2),h),one_one_nat)),
            one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_9304_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(Xa),Xaa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
             => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2,Summary)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2,Summary)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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 = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    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 = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,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 = Mi2,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(TreeList2,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(TreeList2)),
                                              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(TreeList2,h)),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l),
                                                  $let(
                                                    newlist: list_VEBT_VEBT,
                                                    newlist:= list_u1324408373059187874T_VEBT(TreeList2,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),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 = Ma2,
                                                                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),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))))),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 = Ma2,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)))),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_9305_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,M: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),Na) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na) ) ) ).

% set_n_deg_not_0
tff(fact_9306_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A4: $o,B5: $o] : A1 = vEBT_Leaf((A4),(B5))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary) )
             => ( ( A22 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                     => aa_nat_o(vEBT_invar_vebt(X5),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary),M4)
                   => ( ( 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))),M4) )
                     => ( ( M4 = N )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                         => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_13)
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X5),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary) )
               => ( ( A22 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                       => aa_nat_o(vEBT_invar_vebt(X5),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary),M4)
                     => ( ( 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))),M4) )
                       => ( ( M4 = aa_nat_nat(suc,N) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                           => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_13)
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X5),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(vEBT_invar_vebt(X5),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary),M4)
                       => ( ( 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))),M4) )
                         => ( ( M4 = N )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                             => ( ! [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))),M4))
                                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),X_12)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X5),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))),Deg2))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [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))),M4))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                                                & ! [X5: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N) = I4 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X5)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(vEBT_invar_vebt(X5),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary),M4)
                         => ( ( 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))),M4) )
                           => ( ( M4 = aa_nat_nat(suc,N) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                               => ( ! [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))),M4))
                                     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),X_12)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                                         => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X5),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))),Deg2))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [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))),M4))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                                                  & ! [X5: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N) = I4 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X5)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_9307_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info2: option4927543243414619207at_nat,Deg: nat,Summary2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),set_VEBT_VEBT2(TreeList))
     => 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,T))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Info2,Deg,TreeList,Summary2))) ) ).

% height_compose_child
tff(fact_9308_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Deg)
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList))
             => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X5),X_13) )
          & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_9309_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_T_m_a_x_t(vEBT_Leaf((A),(B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_9310_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,vEBT_T_m_a_x_t(T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% maxt_bound
tff(fact_9311_mint__bound,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,vEBT_T_m_i_n_t(T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% mint_bound
tff(fact_9312_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_T_m_i_n_t(vEBT_Leaf((A),(B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_9313_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xa) = Y )
     => ( ! [A4: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B5)) )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((B5),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] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy2,Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_9314_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary2),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_9315_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary2),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Na) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_9316_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xa) = Y )
     => ( ! [A4: $o] :
            ( ? [B5: $o] : Xa = vEBT_Leaf((A4),(B5))
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((A4),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] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy2,Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_9317_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary2),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ! [I3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I3)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Na) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),Na)) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_9318_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa) = 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),Xa),
          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,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              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))),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 )
                        & 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(Summary2,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(Summary2),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)))),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.simps(7)
tff(fact_9319_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa) = 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,Xa),Mi),
          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,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              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),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 )
                          & 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(Summary2,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(Summary2),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)))),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.simps(6)
tff(fact_9320_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary2),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Na) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ! [I3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I3)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Na) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),Na)) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_9321_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(Xa),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,Uw2: $o] : Xa = vEBT_Leaf((A4),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
         => ( ! [A4: $o,B5: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B5)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $ite((B5),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] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V3: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,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))),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                            $ite(
                                              ( ( minlow != none_nat )
                                              & 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(TreeList2,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)))),vEBT_T_m_a_x_t(nth_VEBT_VEBT(TreeList2,the_nat(pr)))))) ) )) ),
                                        one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_9322_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(Xa),Xaa) = Y )
     => ( ( ? [Uu2: $o,B5: $o] : Xa = vEBT_Leaf((Uu2),(B5))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xa = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                              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(TreeList2)),
                                      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),vEBT_T_m_a_x_t(nth_VEBT_VEBT(TreeList2,h)))),
                                        $let(
                                          maxlow: option_nat,
                                          maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,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 )
                                              & 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(TreeList2,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)))),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,the_nat(sc)))))) ) )) )),
                                      one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_9323_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,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)),Xa) = 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,Xa),Mi)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) ),
          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(
              ( ( 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,vEBT_T_m_i_n_t(Summary2)),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),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(Summary2))),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(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),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,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(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),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(Summary2),h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= vEBT_vebt_delete(Summary2,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),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))))),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)))),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_9324_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A8: $o,B8: $o] : A1 = vEBT_Leaf((A8),(B8))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N2) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),N2)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N2) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N2))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = 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,N2)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N2) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),N2)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2) )
            & ! [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))),N2))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I2)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I2) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [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))),N2))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N2) = I2 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N2)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I2 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N2) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N2))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = 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,N2)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) )
            & ! [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N2)))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I2)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I2) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N2)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N2) = I2 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N2)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I2 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_9325_setceilmax,axiom,
    ! [Sa: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Sa),M)
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Listy))
           => aa_nat_o(vEBT_invar_vebt(X),Na) )
       => ( ( M = aa_nat_nat(suc,Na) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Listy))
               => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X)) = 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,Na))) ) )
           => ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Sa)) = 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,M))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Sa,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,M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_9326_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Na: 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,Na),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,Na),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))))) ) ).

% max_idx_list
tff(fact_9327_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,B5: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B5)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B5: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B5)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2,Summary) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2,Summary),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2,Summary) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2,Summary),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                    | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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 = Mi2 )
                                        & ( Xaa = Ma2 ) ),
                                        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 = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,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 = Mi2,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(TreeList2,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(TreeList2)),
                                                  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(TreeList2,h)),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l),
                                                      $let(
                                                        newlist: list_VEBT_VEBT,
                                                        newlist:= list_u1324408373059187874T_VEBT(TreeList2,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),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 = Ma2,
                                                                    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),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))))),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 = Ma2,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)))),vEBT_T_m_a_x_t(nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                                                  one_one_nat ) ) ) )) )) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_9328_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),set_VEBT_VEBT2(TreeList))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList))))) ) ).

% height_compose_list
tff(fact_9329_max__ins__scaled,axiom,
    ! [Na: nat,X14: vEBT_VEBT,M: 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,Na),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),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_9330_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),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))) ) ).

% height_i_max
tff(fact_9331_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B5)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B5)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                       => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                  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(TreeList2)),
                                          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),vEBT_T_m_a_x_t(nth_VEBT_VEBT(TreeList2,h)))),
                                            $let(
                                              maxlow: option_nat,
                                              maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,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 )
                                                  & 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(TreeList2,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)))),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,the_nat(sc)))))) ) )) )),
                                          one_one_nat )) ) ) )) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_9332_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B5: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B5)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite((B5),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve),Xaa)) ) )
                 => ( ! [V3: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh,Vi) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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(TreeList2)),
                                            $let(
                                              minlow: option_nat,
                                              minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,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))),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                                $ite(
                                                  ( ( minlow != none_nat )
                                                  & 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(TreeList2,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)))),vEBT_T_m_a_x_t(nth_VEBT_VEBT(TreeList2,the_nat(pr)))))) ) )) ),
                                            one_one_nat )) ) ) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_9333_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,B5: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B5)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B5: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B5)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2,Summary) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2,Summary),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2,Summary) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2,Summary),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                  one_one_nat,
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    one_one_nat,
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,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(TreeList2,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(TreeList2)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(nth_VEBT_VEBT(TreeList2,h)),l)),
                                              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l)),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary),h),one_one_nat)),
                                            one_one_nat ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_9334_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastAtMost
tff(fact_9335_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9336_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9337_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% atMost_Suc_eq_insert_0
tff(fact_9338_vebt__delete_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = vEBT_Leaf($false,(B5)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,B5: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B5)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf((A4),$false) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B5: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B5)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = vEBT_Leaf((A4),(B5)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary) )
                   => ( ( Y = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry) )
                     => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) )
                       => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,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(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi2,xn,Mi2),
                                        $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(TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,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(TreeList2,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(product_Pair_nat_nat(minn),
                                                          $ite(
                                                            xn = Ma2,
                                                            $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)))))) ),
                                                            Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,sn) ),
                                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                        $ite(xn = Ma2,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)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,Summary) ) ) ),
                                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_9339_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B5)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B5)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                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(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & 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(TreeList2,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 ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_9340_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,zero_zero_nat,Ts2,S2) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts2,S2),Xaa)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,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(TreeList2))
                                    & ~ ( ( Xaa = Mi2 )
                                        | ( Xaa = Ma2 ) ) ),
                                    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(TreeList2,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)))))),vEBT_T_m_i_n_N_u_l_l(nth_VEBT_VEBT(TreeList2,h)))),
                                      $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary),h),one_one_nat)),
                                    one_one_nat ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_9341_vebt__insert_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B5)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B5))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,zero_zero_nat,Ts2,S2) )
               => ( ( Y = vEBT_Node(Info,zero_zero_nat,Ts2,S2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts2,S2),Xaa)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
                 => ( ( Y = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,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)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Xaa,Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,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(TreeList2,h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_9342_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B5: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B5)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve),Xaa)) ) )
                 => ( ! [V3: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh,Vi) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & 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(TreeList2,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 ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_9343_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,zero_zero_nat,Ts2,S2) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts2,S2),Xaa)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,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(TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  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(TreeList2,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(TreeList2,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary),h),one_one_nat)),
                                  one_one_nat ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_9344_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2) )
                 => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              $ite(
                                Xaa = Mi2,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    Xaa = Ma2,
                                    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),Mi2),
                                        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,Ma2),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(TreeList2)),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(TreeList2,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) )) )) )) )) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_9345_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Mi2,
                                zero_zero_nat,
                                $ite(
                                  Xaa = Ma2,
                                  zero_zero_nat,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                    zero_zero_nat,
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                      zero_zero_nat,
                                      $ite(
                                        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa)
                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma2) ),
                                        $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(TreeList2)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList2,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 ) ) ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_9346_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Uu2,Deg2,TreeList2,Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_9347_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Uu,Deg,TreeList,Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_9348_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = vEBT_vebt_insert(vEBT_Leaf((A4),(B5)),Xaa) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa)) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
               => ( ( Y = $ite(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))),Deg2)),Xaa),vEBT_Node(Info,Deg2,TreeList2,Summary),vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList2,Summary),Xaa)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,Deg2,TreeList2,Summary),Xaa)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_9349_vebt__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B5),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),Xaa)) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa))
                       => $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_9350_vebt__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B5),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi2,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_9351_vebt__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B5),$false) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa))
                 => ~ $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_9352_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B5),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_9353_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B5),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_9354_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B5),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S2) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S2),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_9355_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) )
         => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2),Xaa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xaa))
                   => ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2),Xaa))
                     => ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_9356_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2),Xaa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xaa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi2 )
                          | ( Xaa = Ma2 )
                          | $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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2),Xaa)) ) )
               => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_9357_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xaa))
               => ~ ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2),Xaa))
                 => ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 )
                      | $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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_9358_pred__lesseq__max,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $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)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & 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,Deg),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(Summary2,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa),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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_9359_pred__less__length__list,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma)
       => ( 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))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $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)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & 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,Deg),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(Summary2,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa),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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_9360_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
       => ( 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))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $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)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & 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,Deg),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(Summary2,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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_9361_add__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xa)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_9362_mul__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xa)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_9363_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_9364_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_9365_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2),Xa) = $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)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & 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,Deg),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(Summary2,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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_9366_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
        some_nat(Mi),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & 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(TreeList,h),l)),
                  $let(
                    sc: option_nat,
                    sc:= vEBT_vebt_succ(Summary2,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(TreeList,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_9367_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
        some_nat(Ma),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & 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(TreeList,h),l)),
                  $let(
                    pr: option_nat,
                    pr:= vEBT_vebt_pred(Summary2,h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa),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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_9368_vebt__succ_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( ! [Uu2: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((Uu2),(B5)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != $ite((B5),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xa = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            some_nat(Mi2),
                            $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(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & 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(TreeList2,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(TreeList2,the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_9369_vebt__pred_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw2: $o] : Xa = vEBT_Leaf((A4),(Uw2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) )
         => ( ! [A4: $o,B5: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B5)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y != $ite(
                        (B5),
                        some_nat(one_one_nat),
                        $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve)
                 => ( Y != none_nat ) )
               => ( ( ? [V3: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              some_nat(Ma2),
                              $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(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & 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(TreeList2,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,Mi2),Xaa),some_nat(Mi2),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(TreeList2,the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_9370_vebt__succ_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B5)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = $ite((B5),some_nat(one_one_nat),none_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B5)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                some_nat(Mi2),
                                $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(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & 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(TreeList2,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(TreeList2,the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_9371_vebt__pred_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = $ite((A4),some_nat(zero_zero_nat),none_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B5: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B5)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = $ite(
                              (B5),
                              some_nat(one_one_nat),
                              $ite((A4),some_nat(zero_zero_nat),none_nat) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B5)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve),Xaa)) ) )
                 => ( ! [V3: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh,Vi) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  some_nat(Ma2),
                                  $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(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & 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(TreeList2,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,Mi2),Xaa),some_nat(Mi2),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(TreeList2,the_nat(pr)))) ) ) ) ),
                                        none_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_9372_foldr__zero,axiom,
    ! [Xs: list_nat,D: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs),I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),D)),D)) ) ).

% foldr_zero
tff(fact_9373_foldr__one,axiom,
    ! [D: nat,Ys: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ).

% foldr_one
tff(fact_9374_foldr__same__int,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ! [X: nat,Y2: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => ( aa_set_nat_o(member_nat(Y2),set_nat2(Xs))
           => ( X = Y2 ) ) )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
           => ( X = Y ) )
       => ( aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_9375_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Ys),I3)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),C)),size_size_list_nat(Ys))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ) ) ) ).

% foldr_mono
tff(fact_9376_foldr__same,axiom,
    ! [Xs: list_real,Y: real] :
      ( ! [X: real,Y2: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => ( aa_set_real_o(member_real(Y2),set_real2(Xs))
           => ( X = Y2 ) ) )
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_real2(Xs))
           => ( X = Y ) )
       => ( aa_real_real(foldr_real_real(plus_plus_real,Xs),zero_zero_real) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_9377_vebt__delete__code_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] : vEBT_vebt_delete(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa) = case_o2442805151034396888at_nat(vEBT_Node(Info2,Deg,TreeList,Summary2),aa_nat6988169923125987140T_VEBT(aa_VEB5764018861059009915T_VEBT(aa_lis2657662162680668108T_VEBT(aa_nat7935216168507688033T_VEBT(aTP_Lamm_iq(Info2),Deg),TreeList),Summary2),Xa),Info2) ).

% vebt_delete_code(2)
tff(fact_9378_foldr0,axiom,
    ! [Xs: list_real,C: real,D: real] : aa_real_real(foldr_real_real(plus_plus_real,Xs),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs),D)),C) ).

% foldr0
tff(fact_9379_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,vEBT_Node(Info2,Deg,TreeList,Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_9380_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( Y != one_one_nat ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_9381_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,vEBT_Node(Info2,Deg,TreeList,Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space'.simps(2)
tff(fact_9382_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_9383_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,vEBT_Node(Info2,Deg,TreeList,Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space.simps(2)
tff(fact_9384_vebt__insert__code_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Deg),one_one_nat),vEBT_Node(Info2,Deg,TreeList,Summary2),case_o2442805151034396888at_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),Deg,TreeList,Summary2),produc3169358591047799142T_VEBT(aa_nat1426624462703894328T_VEBT(aa_VEB257809287009931607T_VEBT(aa_lis5232481317877780700T_VEBT(aTP_Lamm_ir(Deg),TreeList),Summary2),Xa)),Info2)) ).

% vebt_insert_code(2)
tff(fact_9385_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xa) = Y )
     => ( ( ? [A4: $o,B5: $o] : Xa = vEBT_Leaf((A4),(B5))
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_9386_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_is(L),Qr) ).

% divmod_step_int_def
tff(fact_9387_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_it(L),Qr) ).

% divmod_step_nat_def
tff(fact_9388_vebt__succ__code_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_succ(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Deg),one_one_nat),none_nat,case_o1383228350324149268at_nat(none_nat,produc2484365769952853102on_nat(aa_nat1972073180414880072on_nat(aa_VEB5583160467804174207on_nat(aa_lis5680555565288705232on_nat(aTP_Lamm_iu(Deg),TreeList),Summary2),Xa)),Info2)) ).

% vebt_succ_code(2)
tff(fact_9389_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_iv(L),Qr) ).

% divmod_step_integer_def
tff(fact_9390_divmod__integer_H__def,axiom,
    ! [M: num,Na: num] : unique3479559517661332726nteger(M,Na) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Na)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% divmod_integer'_def
tff(fact_9391_divmod__int__def,axiom,
    ! [M: num,Na: num] : unique5052692396658037445od_int(M,Na) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))) ).

% divmod_int_def
tff(fact_9392_divmod_H__nat__def,axiom,
    ! [M: num,Na: num] : unique5055182867167087721od_nat(M,Na) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))) ).

% divmod'_nat_def
tff(fact_9393_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_9394_abs__integer__code,axiom,
    ! [K: code_integer] :
      abs_abs_Code_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(K),K) ).

% abs_integer_code
tff(fact_9395_minus__one__div__numeral,axiom,
    ! [Na: 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,Na)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Na))) ).

% minus_one_div_numeral
tff(fact_9396_one__div__minus__numeral,axiom,
    ! [Na: 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,Na))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Na))) ).

% one_div_minus_numeral
tff(fact_9397_numeral__div__minus__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Na))) ).

% numeral_div_minus_numeral
tff(fact_9398_minus__numeral__div__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Na))) ).

% minus_numeral_div_numeral
tff(fact_9399_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),
        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:= 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_9400_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(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))),R)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_9401_less__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xb3) ) ).

% less_integer.abs_eq
tff(fact_9402_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: int] : plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xa),aa_int_Code_integer(code_integer_of_int,Xb3)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xb3)) ).

% plus_integer.abs_eq
tff(fact_9403_times__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: int] : times_3573771949741848930nteger(aa_int_Code_integer(code_integer_of_int,Xa),aa_int_Code_integer(code_integer_of_int,Xb3)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xb3)) ).

% times_integer.abs_eq
tff(fact_9404_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: int] : divide6298287555418463151nteger(aa_int_Code_integer(code_integer_of_int,Xa),aa_int_Code_integer(code_integer_of_int,Xb3)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Xb3)) ).

% divide_integer.abs_eq
tff(fact_9405_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: int] : modulo364778990260209775nteger(aa_int_Code_integer(code_integer_of_int,Xa),aa_int_Code_integer(code_integer_of_int,Xb3)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),Xb3)) ).

% modulo_integer.abs_eq
tff(fact_9406_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = R ) ) ).

% mod_int_unique
tff(fact_9407_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_9408_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(product_Pair_int_int(Q2),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_9409_eucl__rel__int,axiom,
    ! [K: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ).

% eucl_rel_int
tff(fact_9410_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R))
    <=> ( ( 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)),R) )
        & $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),R)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R),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),R)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int) ),
              Q2 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_9411_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(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))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_9412_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      divmod_nat(M,Na) = $ite(
        ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
        produc2626176000494625587at_nat(aTP_Lamm_iw,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_9413_divmod__nat__def,axiom,
    ! [M: nat,Na: nat] : divmod_nat(M,Na) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) ).

% divmod_nat_def
tff(fact_9414_sinh__real__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sinh_real,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% sinh_real_zero_iff
tff(fact_9415_sinh__real__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% sinh_real_less_iff
tff(fact_9416_sinh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% sinh_real_le_iff
tff(fact_9417_sinh__real__abs,axiom,
    ! [Xa: real] : aa_real_real(sinh_real,aa_real_real(abs_abs_real,Xa)) = aa_real_real(abs_abs_real,aa_real_real(sinh_real,Xa)) ).

% sinh_real_abs
tff(fact_9418_sinh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_9419_sinh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% sinh_real_pos_iff
tff(fact_9420_sinh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% sinh_real_nonneg_iff
tff(fact_9421_sinh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_9422_arsinh__sinh__real,axiom,
    ! [Xa: real] : aa_real_real(arsinh_real,aa_real_real(sinh_real,Xa)) = Xa ).

% arsinh_sinh_real
tff(fact_9423_sinh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xa),aa_real_real(inverse_inverse_real,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9424_vebt__buildup_Oelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        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_9425_cosh__real__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(cosh_real,Xa) = aa_real_real(cosh_real,Y) )
    <=> ( aa_real_real(abs_abs_real,Xa) = aa_real_real(abs_abs_real,Y) ) ) ).

% cosh_real_eq_iff
tff(fact_9426_cosh__real__abs,axiom,
    ! [Xa: real] : aa_real_real(cosh_real,aa_real_real(abs_abs_real,Xa)) = aa_real_real(cosh_real,Xa) ).

% cosh_real_abs
tff(fact_9427_sinh__real__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(sinh_real,Xa) = aa_real_real(sinh_real,Y) )
    <=> ( Xa = Y ) ) ).

% sinh_real_eq_iff
tff(fact_9428_sinh__less__cosh__real,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),aa_real_real(cosh_real,Xa)) ).

% sinh_less_cosh_real
tff(fact_9429_sinh__le__cosh__real,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),aa_real_real(cosh_real,Xa)) ).

% sinh_le_cosh_real
tff(fact_9430_cosh__real__nonzero,axiom,
    ! [Xa: real] : aa_real_real(cosh_real,Xa) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_9431_cosh__real__pos,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_pos
tff(fact_9432_cosh__real__nonpos__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9433_cosh__real__nonneg__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9434_cosh__real__nonneg,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_nonneg
tff(fact_9435_cosh__real__ge__1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_ge_1
tff(fact_9436_cosh__real__strict__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_9437_cosh__real__nonneg__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9438_cosh__real__nonpos__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9439_arcosh__cosh__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xa)) = Xa ) ) ).

% arcosh_cosh_real
tff(fact_9440_cosh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(inverse_inverse_real,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9441_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_9442_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_9443_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V1022390504157884413omplex(Z) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Z != complex2(aa_real_real(cos_real,T4),aa_real_real(sin_real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9444_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_9445_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_9446_or__minus__numerals_I2_J,axiom,
    ! [Na: 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,Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))) ).

% or_minus_numerals(2)
tff(fact_9447_or__minus__numerals_I6_J,axiom,
    ! [Na: 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,Na)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))) ).

% or_minus_numerals(6)
tff(fact_9448_complex__scaleR,axiom,
    ! [R: real,A: real,B: real] : real_V2046097035970521341omplex(R,complex2(A,B)) = complex2(aa_real_real(times_times_real(R),A),aa_real_real(times_times_real(R),B)) ).

% complex_scaleR
tff(fact_9449_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_9450_OR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),Y)) ) ) ).

% OR_lower
tff(fact_9451_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( complex2(A,B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = aa_num_real(numeral_numeral_real,W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_9452_Complex__mult__complex__of__real,axiom,
    ! [Xa: real,Y: real,R: real] : aa_complex_complex(times_times_complex(complex2(Xa,Y)),real_V4546457046886955230omplex(R)) = complex2(aa_real_real(times_times_real(Xa),R),aa_real_real(times_times_real(Y),R)) ).

% Complex_mult_complex_of_real
tff(fact_9453_complex__of__real__mult__Complex,axiom,
    ! [R: real,Xa: real,Y: real] : aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R)),complex2(Xa,Y)) = complex2(aa_real_real(times_times_real(R),Xa),aa_real_real(times_times_real(R),Y)) ).

% complex_of_real_mult_Complex
tff(fact_9454_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( complex2(A,B) = uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_9455_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(complex2(A,B)),complex2(C,D)) = complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B),C))) ).

% complex_mult
tff(fact_9456_complex__norm,axiom,
    ! [Xa: real,Y: real] : real_V1022390504157884413omplex(complex2(Xa,Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_9457_OR__upper,axiom,
    ! [Xa: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% OR_upper
tff(fact_9458_complex__inverse,axiom,
    ! [A: real,B: real] : invers8013647133539491842omplex(complex2(A,B)) = complex2(aa_real_real(divide_divide_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9459_cot__less__zero,axiom,
    ! [Xa: 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)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xa)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_9460_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_9461_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_9462_or__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_nat_numerals(4)
tff(fact_9463_cot__npi,axiom,
    ! [Na: nat] : aa_real_real(cot_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_9464_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_9465_or__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_nat_numerals(3)
tff(fact_9466_cot__periodic,axiom,
    ! [Xa: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xa) ).

% cot_periodic
tff(fact_9467_divide__complex__def,axiom,
    ! [Xa: complex,Y: complex] : divide1717551699836669952omplex(Xa,Y) = aa_complex_complex(times_times_complex(Xa),invers8013647133539491842omplex(Y)) ).

% divide_complex_def
tff(fact_9468_or__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Na) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% or_nat_def
tff(fact_9469_or__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Na) = $ite(
        M = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9470_cot__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) ) ) ).

% cot_gt_zero
tff(fact_9471_tan__cot_H,axiom,
    ! [Xa: 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)))),Xa)) = aa_real_real(cot_real,Xa) ).

% tan_cot'
tff(fact_9472_exp__two__pi__i,axiom,
    exp_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),real_V4546457046886955230omplex(pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9473_exp__two__pi__i_H,axiom,
    exp_complex(aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9474_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_9475_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_9476_divide__i,axiom,
    ! [Xa: complex] : divide1717551699836669952omplex(Xa,imaginary_unit) = aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(imaginary_unit)),Xa) ).

% divide_i
tff(fact_9477_divide__numeral__i,axiom,
    ! [Z: complex,Na: num] : divide1717551699836669952omplex(Z,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Na)),imaginary_unit)) = divide1717551699836669952omplex(uminus1482373934393186551omplex(aa_complex_complex(times_times_complex(imaginary_unit),Z)),numera6690914467698888265omplex(Na)) ).

% divide_numeral_i
tff(fact_9478_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = uminus1482373934393186551omplex(one_one_complex) ).

% power2_i
tff(fact_9479_i__even__power,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na) ).

% i_even_power
tff(fact_9480_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_9481_semiring__norm_I28_J,axiom,
    ! [Na: num] : aa_num_num(bitM,aa_num_num(bit1,Na)) = aa_num_num(bit1,aa_num_num(bit0,Na)) ).

% semiring_norm(28)
tff(fact_9482_semiring__norm_I27_J,axiom,
    ! [Na: num] : aa_num_num(bitM,aa_num_num(bit0,Na)) = aa_num_num(bit1,aa_num_num(bitM,Na)) ).

% semiring_norm(27)
tff(fact_9483_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Na))) ).

% eval_nat_numeral(2)
tff(fact_9484_BitM__plus__one,axiom,
    ! [Na: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,Na)),one) = aa_num_num(bit0,Na) ).

% BitM_plus_one
tff(fact_9485_one__plus__BitM,axiom,
    ! [Na: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,Na)) = aa_num_num(bit0,Na) ).

% one_plus_BitM
tff(fact_9486_Arg__minus__ii,axiom,
    arg(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_9487_csqrt__ii,axiom,
    csqrt(imaginary_unit) = divide1717551699836669952omplex(plus_plus_complex(one_one_complex,imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9488_Arg__ii,axiom,
    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_9489_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(csqrt(Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_9490_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),arg(Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_9491_cis__minus__pi__half,axiom,
    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))))) = uminus1482373934393186551omplex(imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9492_or__minus__numerals_I1_J,axiom,
    ! [Na: 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,Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,Na)))) ).

% or_minus_numerals(1)
tff(fact_9493_or__minus__numerals_I5_J,axiom,
    ! [Na: 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,Na)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,Na)))) ).

% or_minus_numerals(5)
tff(fact_9494_or__minus__numerals_I4_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bit0,Na)))) ).

% or_minus_numerals(4)
tff(fact_9495_or__minus__numerals_I8_J,axiom,
    ! [Na: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bit0,Na)))) ).

% or_minus_numerals(8)
tff(fact_9496_or__minus__numerals_I7_J,axiom,
    ! [Na: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bitM,Na)))) ).

% or_minus_numerals(7)
tff(fact_9497_or__minus__numerals_I3_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bitM,Na)))) ).

% or_minus_numerals(3)
tff(fact_9498_cis__pi__half,axiom,
    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_9499_cis__2pi,axiom,
    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_9500_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(bit_or_not_num_neg(one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9501_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Na: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Na)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_9502_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Na: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Na)),aa_num_num(bit1,M)) = aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(Na),M)) ).

% or_not_num_neg.simps(6)
tff(fact_9503_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Na: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Na)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9504_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_9505_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Na: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Na)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Na),M)) ).

% or_not_num_neg.simps(5)
tff(fact_9506_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Na: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Na)),aa_num_num(bit1,M)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Na),M)) ).

% or_not_num_neg.simps(9)
tff(fact_9507_cis__divide,axiom,
    ! [A: real,B: real] : divide1717551699836669952omplex(cis(A),cis(B)) = cis(aa_real_real(minus_minus_real(A),B)) ).

% cis_divide
tff(fact_9508_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit0,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_9509_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Na: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Na)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Na),M)) ).

% or_not_num_neg.simps(8)
tff(fact_9510_DeMoivre,axiom,
    ! [A: real,Na: nat] : aa_nat_complex(power_power_complex(cis(A)),Na) = cis(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),A)) ).

% DeMoivre
tff(fact_9511_or__not__num__neg_Oelims,axiom,
    ! [Xa: num,Xaa: num,Y: num] :
      ( ( aa_num_num(bit_or_not_num_neg(Xa),Xaa) = Y )
     => ( ( ( Xa = one )
         => ( ( Xaa = one )
           => ( Y != one ) ) )
       => ( ( ( Xa = one )
           => ! [M4: num] :
                ( ( Xaa = aa_num_num(bit0,M4) )
               => ( Y != aa_num_num(bit1,M4) ) ) )
         => ( ( ( Xa = one )
             => ! [M4: num] :
                  ( ( Xaa = aa_num_num(bit1,M4) )
                 => ( Y != aa_num_num(bit1,M4) ) ) )
           => ( ( ? [N: num] : Xa = aa_num_num(bit0,N)
               => ( ( Xaa = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( Xa = aa_num_num(bit0,N) )
                   => ! [M4: num] :
                        ( ( Xaa = aa_num_num(bit0,M4) )
                       => ( Y != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M4)) ) ) )
               => ( ! [N: num] :
                      ( ( Xa = aa_num_num(bit0,N) )
                     => ! [M4: num] :
                          ( ( Xaa = aa_num_num(bit1,M4) )
                         => ( Y != aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N),M4)) ) ) )
                 => ( ( ? [N: num] : Xa = aa_num_num(bit1,N)
                     => ( ( Xaa = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( Xa = aa_num_num(bit1,N) )
                         => ! [M4: num] :
                              ( ( Xaa = aa_num_num(bit0,M4) )
                             => ( Y != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M4)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xa = aa_num_num(bit1,N) )
                           => ! [M4: num] :
                                ( ( Xaa = aa_num_num(bit1,M4) )
                               => ( Y != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9512_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_ix(Na),aa_nat_set_nat(set_ord_lessThan_nat,Na)),collect_complex(aTP_Lamm_gz(Na))) ) ).

% bij_betw_roots_unity
tff(fact_9513_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_iy(D))) ).

% int_ge_less_than_def
tff(fact_9514_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_iz(D))) ).

% int_ge_less_than2_def
tff(fact_9515_set__decode__0,axiom,
    ! [Xa: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xa))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xa) ) ).

% set_decode_0
tff(fact_9516_bij__betw__Suc,axiom,
    ! [M7: set_nat,N3: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M7),N3)
    <=> ( image_nat_nat(suc,M7) = N3 ) ) ).

% bij_betw_Suc
tff(fact_9517_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_9518_set__decode__Suc,axiom,
    ! [Na: nat,Xa: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Na)),aa_nat_set_nat(nat_set_decode,Xa))
    <=> aa_set_nat_o(member_nat(Na),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_9519_set__decode__plus__power__2,axiom,
    ! [Na: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Na),aa_nat_set_nat(nat_set_decode,Z))
     => ( aa_nat_set_nat(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))),Na)),Z)) = insert_nat(Na,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_9520_set__decode__def,axiom,
    ! [Xa: nat] : aa_nat_set_nat(nat_set_decode,Xa) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ja,Xa)) ).

% set_decode_def
tff(fact_9521_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Na: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(aa_real_real(aa_nat_fun_real_real(root,Na),real_V1022390504157884413omplex(C)))),cis(aa_real_real(divide_divide_real(arg(C)),aa_nat_real(semiri5074537144036343181t_real,Na))))),collect_complex(aTP_Lamm_gz(Na))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_jb(C),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9522_forall__finite_I3_J,axiom,
    ! [Xa: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(suc,Xa)))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Xa))
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% forall_finite(3)
tff(fact_9523_forall__finite_I2_J,axiom,
    ! [P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,zero_zero_nat))
         => aa_nat_o(P,I2) )
    <=> aa_nat_o(P,zero_zero_nat) ) ).

% forall_finite(2)
tff(fact_9524_real__root__zero,axiom,
    ! [Na: nat] : aa_real_real(aa_nat_fun_real_real(root,Na),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9525_real__root__Suc__0,axiom,
    ! [Xa: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xa) = Xa ).

% real_root_Suc_0
tff(fact_9526_real__root__eq__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = aa_real_real(aa_nat_fun_real_real(root,Na),Y) )
      <=> ( Xa = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9527_root__0,axiom,
    ! [Xa: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xa) = zero_zero_real ).

% root_0
tff(fact_9528_real__root__eq__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = zero_zero_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9529_real__root__less__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ).

% real_root_less_iff
tff(fact_9530_real__root__le__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% real_root_le_iff
tff(fact_9531_real__root__eq__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = one_one_real )
      <=> ( Xa = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9532_real__root__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9533_real__root__lt__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9534_real__root__gt__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_9535_real__root__le__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9536_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_9537_real__root__lt__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9538_real__root__gt__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_9539_real__root__le__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9540_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_9541_real__root__pow__pos2,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos2
tff(fact_9542_real__root__divide,axiom,
    ! [Na: nat,Xa: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(divide_divide_real(Xa),Y)) = aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ).

% real_root_divide
tff(fact_9543_real__root__commute,axiom,
    ! [M: nat,Na: nat,Xa: real] : aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) = aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(aa_nat_fun_real_real(root,M),Xa)) ).

% real_root_commute
tff(fact_9544_real__root__minus,axiom,
    ! [Na: nat,Xa: real] : aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ).

% real_root_minus
tff(fact_9545_real__root__inverse,axiom,
    ! [Na: nat,Xa: real] : aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ).

% real_root_inverse
tff(fact_9546_real__root__mult,axiom,
    ! [Na: nat,Xa: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(times_times_real(Xa),Y)) = aa_real_real(times_times_real(aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ).

% real_root_mult
tff(fact_9547_real__root__mult__exp,axiom,
    ! [M: nat,Na: nat,Xa: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),Xa) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ).

% real_root_mult_exp
tff(fact_9548_real__root__pos__pos__le,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ).

% real_root_pos_pos_le
tff(fact_9549_real__root__less__mono,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_less_mono
tff(fact_9550_real__root__le__mono,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_9551_real__root__power,axiom,
    ! [Na: nat,Xa: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),K) ) ) ).

% real_root_power
tff(fact_9552_real__root__abs,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(abs_abs_real,Xa)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ).

% real_root_abs
tff(fact_9553_real__root__gt__zero,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ).

% real_root_gt_zero
tff(fact_9554_real__root__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N3),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9555_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_9556_root__abs__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9557_real__root__pos__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ).

% real_root_pos_pos
tff(fact_9558_real__root__strict__increasing,axiom,
    ! [Na: nat,N3: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,N3),Xa)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9559_real__root__decreasing,axiom,
    ! [Na: nat,N3: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N3),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ) ).

% real_root_decreasing
tff(fact_9560_odd__real__root__power__cancel,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = Xa ) ) ).

% odd_real_root_power_cancel
tff(fact_9561_odd__real__root__unique,axiom,
    ! [Na: nat,Y: real,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na) = Xa )
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9562_odd__real__root__pow,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),Na) = Xa ) ) ).

% odd_real_root_pow
tff(fact_9563_real__root__pow__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos
tff(fact_9564_real__root__power__cancel,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = Xa ) ) ) ).

% real_root_power_cancel
tff(fact_9565_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na) = Xa )
         => ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9566_real__root__increasing,axiom,
    ! [Na: nat,N3: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,N3),Xa)) ) ) ) ) ).

% real_root_increasing
tff(fact_9567_log__root,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(log(B),aa_real_real(aa_nat_fun_real_real(root,Na),A)) = aa_real_real(divide_divide_real(aa_real_real(log(B),A)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_root
tff(fact_9568_log__base__root,axiom,
    ! [Na: nat,B: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( 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,Na),B)),Xa) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(B),Xa)) ) ) ) ).

% log_base_root
tff(fact_9569_ln__root,axiom,
    ! [Na: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( 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,Na),B)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% ln_root
tff(fact_9570_forall__finite_I1_J,axiom,
    ! [P: fun_nat_o,I4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),zero_zero_nat)
     => aa_nat_o(P,I4) ) ).

% forall_finite(1)
tff(fact_9571_root__powr__inverse,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = aa_real_real(powr_real(Xa),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ) ).

% root_powr_inverse
tff(fact_9572_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xa: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Xa))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Xa)
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_9573_uint32_Osize__eq,axiom,
    ! [P3: uint32] : size_size_uint32(P3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) ).

% uint32.size_eq
tff(fact_9574_nat__of__bool,axiom,
    ! [P: $o] : aa_int_nat(nat2,aa_o_int(zero_n2684676970156552555ol_int,(P))) = zero_n2687167440665602831ol_nat((P)) ).

% nat_of_bool
tff(fact_9575_Suc__0__mod__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = zero_n2687167440665602831ol_nat(Na != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_9576_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : adjust_div(aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_o_int(zero_n2684676970156552555ol_int,R != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_9577_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_jc,Qr) ).

% Divides.adjust_div_def
tff(fact_9578_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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))),Na))) ).

% Suc_0_or_eq
tff(fact_9579_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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))),Na))) ).

% or_Suc_0_eq
tff(fact_9580_or__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9581_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_9582_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_9583_and__int_Oelims,axiom,
    ! [Xa: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Xaa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_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))),Xa)
                & ~ 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))),Xa)
                  & ~ 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,Xa),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_9584_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_9585_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_9586_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_9587_and__minus__numerals_I6_J,axiom,
    ! [Na: 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,Na)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_9588_and__minus__numerals_I2_J,axiom,
    ! [Na: 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,Na)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_9589_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_Suc_0
tff(fact_9590_and__minus__numerals_I5_J,axiom,
    ! [Na: 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,Na)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_9591_and__minus__numerals_I1_J,axiom,
    ! [Na: 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,Na)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_9592_take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : bit_se2923211474154528505it_int(Na,aa_int_int(aa_int_fun_int_int(times_times_int,bit_se2923211474154528505it_int(Na,K)),bit_se2923211474154528505it_int(Na,L))) = bit_se2923211474154528505it_int(Na,aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% take_bit_mult
tff(fact_9593_take__bit__minus,axiom,
    ! [Na: nat,K: int] : bit_se2923211474154528505it_int(Na,aa_int_int(uminus_uminus_int,bit_se2923211474154528505it_int(Na,K))) = bit_se2923211474154528505it_int(Na,aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_9594_take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : bit_se2923211474154528505it_int(Na,aa_int_int(aa_int_fun_int_int(minus_minus_int,bit_se2923211474154528505it_int(Na,K)),bit_se2923211474154528505it_int(Na,L))) = bit_se2923211474154528505it_int(Na,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_9595_take__bit__nat__eq,axiom,
    ! [K: int,Na: 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,Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,bit_se2923211474154528505it_int(Na,K)) ) ) ).

% take_bit_nat_eq
tff(fact_9596_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,bit_se2923211474154528505it_int(Na,K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9597_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9598_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_9599_num__induct,axiom,
    ! [P: fun_num_o,Xa: num] :
      ( aa_num_o(P,one)
     => ( ! [X: num] :
            ( aa_num_o(P,X)
           => aa_num_o(P,aa_num_num(inc,X)) )
       => aa_num_o(P,Xa) ) ) ).

% num_induct
tff(fact_9600_add__inc,axiom,
    ! [Xa: num,Y: num] : aa_num_num(plus_plus_num(Xa),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(plus_plus_num(Xa),Y)) ).

% add_inc
tff(fact_9601_AND__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => 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,Xa),Y)) ) ).

% AND_lower
tff(fact_9602_AND__upper1,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)),Xa) ) ).

% AND_upper1
tff(fact_9603_AND__upper2,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)),Y) ) ).

% AND_upper2
tff(fact_9604_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_9605_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_9606_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se2923211474154528505it_int(M,K)),bit_se2923211474154528505it_int(Na,K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9607_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se2923211474154528505it_int(Na,K)) ).

% take_bit_nonnegative
tff(fact_9608_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se2923211474154528505it_int(Na,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_9609_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se2923211474154528505it_int(Na,K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_9610_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),bit_se2923211474154528505it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_9611_plus__and__or,axiom,
    ! [Xa: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y) ).

% plus_and_or
tff(fact_9612_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9613_inc_Osimps_I2_J,axiom,
    ! [Xa: num] : aa_num_num(inc,aa_num_num(bit0,Xa)) = aa_num_num(bit1,Xa) ).

% inc.simps(2)
tff(fact_9614_inc_Osimps_I3_J,axiom,
    ! [Xa: num] : aa_num_num(inc,aa_num_num(bit1,Xa)) = aa_num_num(bit0,aa_num_num(inc,Xa)) ).

% inc.simps(3)
tff(fact_9615_add__One,axiom,
    ! [Xa: num] : aa_num_num(plus_plus_num(Xa),one) = aa_num_num(inc,Xa) ).

% add_One
tff(fact_9616_inc__BitM__eq,axiom,
    ! [Na: num] : aa_num_num(inc,aa_num_num(bitM,Na)) = aa_num_num(bit0,Na) ).

% inc_BitM_eq
tff(fact_9617_BitM__inc__eq,axiom,
    ! [Na: num] : aa_num_num(bitM,aa_num_num(inc,Na)) = aa_num_num(bit1,Na) ).

% BitM_inc_eq
tff(fact_9618_mult__inc,axiom,
    ! [Xa: num,Y: num] : aa_num_num(times_times_num(Xa),aa_num_num(inc,Y)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(Xa),Y)),Xa) ).

% mult_inc
tff(fact_9619_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_9620_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_9621_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_9622_take__bit__decr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) != zero_zero_int )
     => ( bit_se2923211474154528505it_int(Na,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,bit_se2923211474154528505it_int(Na,K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_9623_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_9624_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9625_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_nat_less_exp
tff(fact_9626_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_9627_take__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_nat_def
tff(fact_9628_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),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,bit_se2923211474154528505it_int(Na,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_9629_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se2923211474154528505it_int(Na,K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_int_less_exp
tff(fact_9630_take__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se2923211474154528505it_int(Na,K) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_int_def
tff(fact_9631_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : 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,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_9632_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_9633_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),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,bit_se2923211474154528505it_int(Na,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_9634_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se2923211474154528505it_int(Na,K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_9635_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),bit_se2923211474154528505it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9636_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_9637_take__bit__int__eq__self,axiom,
    ! [K: int,Na: 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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( bit_se2923211474154528505it_int(Na,K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_9638_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9639_take__bit__incr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int) )
     => ( bit_se2923211474154528505it_int(Na,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),bit_se2923211474154528505it_int(Na,K)) ) ) ).

% take_bit_incr_eq
tff(fact_9640_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : 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,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_9641_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se2923211474154528505it_int(Na,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_9642_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: 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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),bit_se2923211474154528505it_int(Na,K)) ) ).

% take_bit_int_greater_eq
tff(fact_9643_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : bit_ri631733984087533419it_int(Na,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_9644_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: 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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( bit_se2923211474154528505it_int(Na,aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9645_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_9646_and__int_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(Xa),Xaa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_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))),Xa)
                      & ~ 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))),Xa)
                        & ~ 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,Xa),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(product_Pair_int_int(Xa),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_9647_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_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_9648_and__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_9649_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_9650_and__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_9651_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_9652_Suc__0__and__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9653_and__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9654_and__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% and_nat_def
tff(fact_9655_and__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na) = $ite(
        ( ( M = zero_zero_nat )
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_9656_and__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_9657_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(product_Pair_int_int(A0),A1))
     => ( ! [K2: int,L2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K2),L2))
           => ( ( ~ ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9658_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(product_Pair_int_int(A0),A1))
     => ( ! [I3: int,J3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(I3),J3))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J3)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J3) )
             => aa_int_o(aa_int_fun_int_o(P,I3),J3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_9659_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_jd,Y)) ).

% arctan_def
tff(fact_9660_ln__real__def,axiom,
    ! [Xa: real] : aa_real_real(ln_ln_real,Xa) = the_real(aa_real_fun_real_o(aTP_Lamm_je,Xa)) ).

% ln_real_def
tff(fact_9661_ln__neg__is__const,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xa) = the_real(aTP_Lamm_an) ) ) ).

% ln_neg_is_const
tff(fact_9662_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_jf,Y)) ).

% arccos_def
tff(fact_9663_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_jg) ).

% pi_half
tff(fact_9664_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_jg)) ).

% pi_def
tff(fact_9665_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_jh,Y)) ).

% arcsin_def
tff(fact_9666_floor__real__def,axiom,
    ! [Xa: real] : aa_real_int(archim6058952711729229775r_real,Xa) = the_int(aTP_Lamm_ji(Xa)) ).

% floor_real_def
tff(fact_9667_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(sgn_sgn_int(K) = sgn_sgn_int(L),aa_int_int(aa_int_fun_int_int(times_times_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,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_9668_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(sgn_sgn_int(K) = 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)))),zero_n2687167440665602831ol_nat(~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_9669_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: 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),sgn_sgn_int(R)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9670_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: 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,sgn_sgn_int(R)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9671_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: 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),sgn_sgn_int(R))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9672_sgn__mult__dvd__iff,axiom,
    ! [R: 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,sgn_sgn_int(R)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9673_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L2: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(L2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_9674_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( sgn_sgn_int(K) = sgn_sgn_int(L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_9675_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
       => ( sgn_sgn_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) = sgn_sgn_int(L) ) ) ) ).

% sgn_mod
tff(fact_9676_zsgn__def,axiom,
    ! [I: 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_9677_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,sgn_sgn_int(V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_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_9678_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,sgn_sgn_int(K)),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_9679_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( sgn_sgn_int(R) = sgn_sgn_int(L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R)),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)),R) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9680_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( sgn_sgn_int(K) != sgn_sgn_int(L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9681_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(product_Pair_int_int(zero_zero_int),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R4: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R4) )
               => ( ( sgn_sgn_int(R4) = sgn_sgn_int(A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R4)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9682_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K3) ) )
        | ? [L3: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R5) )
            & ( sgn_sgn_int(R5) = sgn_sgn_int(L3) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L3))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9683_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(L)),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( sgn_sgn_int(L) = zero_zero_int )
        | ( sgn_sgn_int(K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        zero_zero_int,
        $ite(sgn_sgn_int(K) = sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)))))) ) ).

% divide_int_unfold
tff(fact_9684_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(L)),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( sgn_sgn_int(L) = zero_zero_int )
        | ( sgn_sgn_int(K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(K)),aa_nat_int(semiri1314217659103216013at_int,M)),
        $ite(sgn_sgn_int(K) = sgn_sgn_int(L),aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))),aa_int_int(aa_int_fun_int_int(times_times_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,Na),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))))) ) ).

% modulo_int_unfold
tff(fact_9685_floor__rat__def,axiom,
    ! [Xa: rat] : aa_rat_int(archim3151403230148437115or_rat,Xa) = the_int(aTP_Lamm_jj(Xa)) ).

% floor_rat_def
tff(fact_9686_sgn__div__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) != zero_zero_int )
     => ( sgn_sgn_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) = sgn_sgn_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_9687_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% mask_nat_positive_iff
tff(fact_9688_real__sgn__eq,axiom,
    ! [Xa: real] : aa_real_real(sgn_sgn_real,Xa) = aa_real_real(divide_divide_real(Xa),aa_real_real(abs_abs_real,Xa)) ).

% real_sgn_eq
tff(fact_9689_less__eq__mask,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ).

% less_eq_mask
tff(fact_9690_nat__mask__eq,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Na)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Na) ).

% nat_mask_eq
tff(fact_9691_sgn__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(sgn_sgn_rat,A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9692_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R)
     => ~ ! [S2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S2)
           => ! [T4: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T4)
               => ( R != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,S2),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9693_abs__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_rat_def
tff(fact_9694_less__eq__rat__def,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9695_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Na)) ).

% mask_nonnegative_int
tff(fact_9696_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_9697_sgn__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) = aa_real_real(sgn_sgn_real,Xa) ) ) ).

% sgn_root
tff(fact_9698_less__mask,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ) ).

% less_mask
tff(fact_9699_sgn__eq,axiom,
    ! [Z: complex] : sgn_sgn_complex(Z) = divide1717551699836669952omplex(Z,real_V4546457046886955230omplex(real_V1022390504157884413omplex(Z))) ).

% sgn_eq
tff(fact_9700_sgn__real__def,axiom,
    ! [A: real] :
      aa_real_real(sgn_sgn_real,A) = $ite(
        A = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_9701_take__bit__eq__mask__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) = aa_nat_int(bit_se2000444600071755411sk_int,Na) )
    <=> ( bit_se2923211474154528505it_int(Na,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_9702_sgn__integer__code,axiom,
    ! [K: code_integer] :
      sgn_sgn_Code_integer(K) = $ite(
        K = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_9703_sgn__power__injE,axiom,
    ! [A: real,Na: nat,Xa: real,B: real] :
      ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na)) = Xa )
     => ( ( Xa = 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)),Na)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9704_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% Suc_mask_eq_exp
tff(fact_9705_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% mask_nat_less_exp
tff(fact_9706_root__sgn__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),Na))) = Y ) ) ).

% root_sgn_power
tff(fact_9707_sgn__power__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa))),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,Na),Xa))),Na)) = Xa ) ) ).

% sgn_power_root
tff(fact_9708_mask__half__int,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),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,Na),one_one_nat)) ).

% mask_half_int
tff(fact_9709_mask__int__def,axiom,
    ! [Na: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Na) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int) ).

% mask_int_def
tff(fact_9710_mask__nat__def,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Na) = 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))),Na)),one_one_nat) ).

% mask_nat_def
tff(fact_9711_cis__Arg__unique,axiom,
    ! [Z: complex,Xa: real] :
      ( ( sgn_sgn_complex(Z) = cis(Xa) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => ( arg(Z) = Xa ) ) ) ) ).

% cis_Arg_unique
tff(fact_9712_split__root,axiom,
    ! [P: fun_real_o,Na: nat,Xa: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Na),Xa))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ! [Y6: real] :
              ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y6)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y6)),Na)) = Xa )
             => aa_real_o(P,Y6) ) ) ) ) ).

% split_root
tff(fact_9713_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( sgn_sgn_complex(Z) = cis(arg(Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),arg(Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_9714_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) = aa_nat_int(bit_se2000444600071755411sk_int,Na) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9715_arctan__inverse,axiom,
    ! [Xa: real] :
      ( ( Xa != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),Xa)) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Xa)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,Xa)) ) ) ).

% arctan_inverse
tff(fact_9716_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : bit_ri631733984087533419it_int(Na,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Na)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9717_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : aa_num_nat(size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_9718_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_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_9719_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_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_9720_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_ri631733984087533419it_int(Na,K))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9721_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_ri631733984087533419it_int(Na,K)),zero_zero_int)
    <=> aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_9722_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa_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,Na))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Na) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9723_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa_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,Na))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),Na) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9724_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Na: num] :
      ( aa_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,Na))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Na)) ) ).

% bit_minus_numeral_int(1)
tff(fact_9725_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Na: num] :
      ( aa_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,Na))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Na)) ) ).

% bit_minus_numeral_int(2)
tff(fact_9726_bin__nth__minus__Bit0,axiom,
    ! [Na: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),Na)
      <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% bin_nth_minus_Bit0
tff(fact_9727_bin__nth__minus__Bit1,axiom,
    ! [Na: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),Na)
      <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% bin_nth_minus_Bit1
tff(fact_9728_and__minus__minus__numerals,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = bit_ri7919022796975470100ot_int(aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Na)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9729_or__minus__minus__numerals,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = bit_ri7919022796975470100ot_int(aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Na)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9730_bit__minus__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K)),Na)
    <=> aa_nat_o(bit_se1146084159140164899it_int(bit_ri7919022796975470100ot_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),Na) ) ).

% bit_minus_int_iff
tff(fact_9731_bit__not__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(bit_ri7919022796975470100ot_int(K)),Na)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% bit_not_int_iff
tff(fact_9732_bit__or__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),Na)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
        | aa_nat_o(bit_se1146084159140164899it_int(L),Na) ) ) ).

% bit_or_int_iff
tff(fact_9733_bit__and__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),Na)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
        & aa_nat_o(bit_se1146084159140164899it_int(L),Na) ) ) ).

% bit_and_int_iff
tff(fact_9734_or__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = bit_ri7919022796975470100ot_int(aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,bit_ri7919022796975470100ot_int(K)),bit_ri7919022796975470100ot_int(L))) ).

% or_int_def
tff(fact_9735_not__int__def,axiom,
    ! [K: 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_9736_and__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),bit_ri7919022796975470100ot_int(one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9737_or__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),bit_ri7919022796975470100ot_int(one_one_int)) = bit_ri7919022796975470100ot_int(zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9738_bit__not__int__iff_H,axiom,
    ! [K: int,Na: nat] :
      ( aa_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)),Na)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% bit_not_int_iff'
tff(fact_9739_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,bit_ri7919022796975470100ot_int(K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = 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_9740_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))),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_9741_and__not__numerals_I2_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9742_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),bit_ri7919022796975470100ot_int(one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(4)
tff(fact_9743_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),bit_se2923211474154528505it_int(M,K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9744_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),bit_ri7919022796975470100ot_int(one_one_int)) = bit_ri7919022796975470100ot_int(one_one_int) ).

% or_not_numerals(4)
tff(fact_9745_or__not__numerals_I2_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))) = bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))) ).

% or_not_numerals(2)
tff(fact_9746_numeral__or__not__num__eq,axiom,
    ! [M: num,Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),Na)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% numeral_or_not_num_eq
tff(fact_9747_int__numeral__not__or__num__neg,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Na),M))) ).

% int_numeral_not_or_num_neg
tff(fact_9748_int__numeral__or__not__num__neg,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),Na))) ).

% int_numeral_or_not_num_neg
tff(fact_9749_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),M2)
              <=> aa_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(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),N) ) ) ) ).

% int_bit_bound
tff(fact_9750_and__not__numerals_I5_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% and_not_numerals(5)
tff(fact_9751_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),bit_ri7919022796975470100ot_int(one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(7)
tff(fact_9752_or__not__numerals_I3_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))) = bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))) ).

% or_not_numerals(3)
tff(fact_9753_and__not__numerals_I3_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9754_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),bit_ri7919022796975470100ot_int(one_one_int)) = bit_ri7919022796975470100ot_int(zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9755_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9756_and__not__numerals_I6_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% and_not_numerals(6)
tff(fact_9757_and__not__numerals_I9_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% and_not_numerals(9)
tff(fact_9758_or__not__numerals_I6_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% or_not_numerals(6)
tff(fact_9759_bit__int__def,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
    <=> ~ 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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ).

% bit_int_def
tff(fact_9760_or__not__numerals_I5_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))))) ).

% or_not_numerals(5)
tff(fact_9761_and__not__numerals_I8_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))))) ).

% and_not_numerals(8)
tff(fact_9762_or__not__numerals_I9_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))))) ).

% or_not_numerals(9)
tff(fact_9763_or__not__numerals_I8_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))))) ).

% or_not_numerals(8)
tff(fact_9764_not__int__rec,axiom,
    ! [K: 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))),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_9765_Bit__Operations_Oset__bit__eq,axiom,
    ! [Na: nat,K: int] : bit_se7879613467334960850it_int(Na,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(bit_se1146084159140164899it_int(K),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ).

% Bit_Operations.set_bit_eq
tff(fact_9766_unset__bit__eq,axiom,
    ! [Na: nat,K: int] : bit_se4203085406695923979it_int(Na,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(bit_se1146084159140164899it_int(K),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ).

% unset_bit_eq
tff(fact_9767_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Na)))),bit_se2923211474154528505it_int(Na,K)) ).

% take_bit_Suc_from_most
tff(fact_9768_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_9769_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),
        bit_ri7919022796975470100ot_int(L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_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_9770_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_9771_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_9772_XOR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa),Y)) ) ) ).

% XOR_lower
tff(fact_9773_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9774_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_9775_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),Na)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
        <=> aa_nat_o(bit_se1146084159140164899it_int(L),Na) ) ) ).

% bit_xor_int_iff
tff(fact_9776_not__bit__Suc__0__numeral,axiom,
    ! [Na: 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,Na)) ).

% not_bit_Suc_0_numeral
tff(fact_9777_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),bit_ri7919022796975470100ot_int(L))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,bit_ri7919022796975470100ot_int(K)),L)) ).

% xor_int_def
tff(fact_9778_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Na)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ) ).

% bit_nat_iff
tff(fact_9779_bit__nat__def,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,M),Na)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% bit_nat_def
tff(fact_9780_XOR__upper,axiom,
    ! [Xa: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% XOR_upper
tff(fact_9781_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_9782_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_jk(Z))) ).

% Arg_def
tff(fact_9783_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = one_one_int )
               => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_9784_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_9785_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_9786_xor__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% xor_nat_numerals(3)
tff(fact_9787_xor__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa)) ).

% xor_nat_numerals(4)
tff(fact_9788_xor__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% xor_nat_def
tff(fact_9789_xor__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na) = $ite(
        M = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9790_xor__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9791_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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))),Na)))),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))),Na))) ).

% Suc_0_xor_eq
tff(fact_9792_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Na),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,Na),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))),Na)))),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))),Na))) ).

% xor_Suc_0_eq
tff(fact_9793_vebt__buildup_Opelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            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_9794_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
             => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_9795_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_9796_VEBT__internal_OTb_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_9797_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_9798_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
             => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
               => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_9799_cis__multiple__2pi,axiom,
    ! [Na: real] :
      ( aa_set_real_o(member_real(Na),ring_1_Ints_real)
     => ( 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)),Na)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9800_sin__times__pi__eq__0,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,aa_real_real(times_times_real(Xa),pi)) = zero_zero_real )
    <=> aa_set_real_o(member_real(Xa),ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9801_sin__integer__2pi,axiom,
    ! [Na: real] :
      ( aa_set_real_o(member_real(Na),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)),Na)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9802_cos__integer__2pi,axiom,
    ! [Na: real] :
      ( aa_set_real_o(member_real(Na),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)),Na)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9803_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_jl,quotient_of(P3)) ).

% rat_inverse_code
tff(fact_9804_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9805_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(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_9806_divide__rat__def,axiom,
    ! [Q2: rat,R: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Q2),R) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Q2),aa_rat_rat(inverse_inverse_rat,R)) ).

% divide_rat_def
tff(fact_9807_quotient__of__div,axiom,
    ! [R: rat,Na: int,D: int] :
      ( ( quotient_of(R) = aa_int7462565939943104719nt_int(product_Pair_int_int(Na),D) )
     => ( R = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_int_rat(ring_1_of_int_rat,Na)),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_9808_quotient__of__denom__pos,axiom,
    ! [R: rat,P3: int,Q2: int] :
      ( ( quotient_of(R) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% quotient_of_denom_pos
tff(fact_9809_rat__uminus__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_jm,quotient_of(P3)) ).

% rat_uminus_code
tff(fact_9810_rat__less__code,axiom,
    ! [P3: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P3),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_jo(Q2)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_9811_rat__floor__code,axiom,
    ! [P3: rat] : aa_rat_int(archim3151403230148437115or_rat,P3) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P3)) ).

% rat_floor_code
tff(fact_9812_rat__abs__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_jp,quotient_of(P3)) ).

% rat_abs_code
tff(fact_9813_rat__less__eq__code,axiom,
    ! [P3: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_jr(Q2)),quotient_of(P3)) ) ).

% rat_less_eq_code
tff(fact_9814_rat__plus__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P3),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_jt(Q2),quotient_of(P3)) ).

% rat_plus_code
tff(fact_9815_normalize__negative,axiom,
    ! [Q2: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q2),zero_zero_int)
     => ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3)),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_9816_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P3: int,Q2: int] :
      ( ( normalize(R) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% normalize_denom_pos
tff(fact_9817_normalize__crossproduct,axiom,
    ! [Q2: int,Sa: int,P3: int,R: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( Sa != zero_zero_int )
       => ( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(R),Sa)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P3),Sa) = aa_int_int(aa_int_fun_int_int(times_times_int,R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9818_rat__divide__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_jv(Q2),quotient_of(P3)) ).

% rat_divide_code
tff(fact_9819_rat__times__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P3),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_jx(Q2),quotient_of(P3)) ).

% rat_times_code
tff(fact_9820_rat__minus__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,P3),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_jz(Q2),quotient_of(P3)) ).

% rat_minus_code
tff(fact_9821_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9822_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_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_9823_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9824_swappa,axiom,
    ! [B2: assn,A2: assn,C2: assn,X7: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),A2)),C2),X7)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),C2),X7) ) ).

% swappa
tff(fact_9825_midextr,axiom,
    ! [P: assn,Q: assn,Q6: assn,R2: assn,X7: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),Q6)),R2),X7)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R2)),Q)),Q6),X7) ) ).

% midextr
tff(fact_9826_groupy,axiom,
    ! [A2: assn,B2: assn,C2: assn,D3: assn,X7: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),D3)),X7)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),C2)),D3),X7) ) ).

% groupy
tff(fact_9827_ent__false,axiom,
    ! [P: assn] : entails(bot_bot_assn,P) ).

% ent_false
tff(fact_9828_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] : entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ).

% assnle
tff(fact_9829_big__assn__simp,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,L: nat,Xa: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),Xa)),aa_o_assn(pure_assn,Xaa = vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList))),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),aa_o_assn(pure_assn,Xaa = vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),list_u6098035379799741383_VEBTi(Tree_is,H,Xa)))) ) ).

% big_assn_simp
tff(fact_9830_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList))),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList))),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y)))) ) ).

% local.ext
tff(fact_9831_atLeastLessThan__singleton,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9832_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList))),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% txe
tff(fact_9833_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,I)),nth_VEBT_VEBTi(Tree_is,I))),Rest)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList))),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Rest),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% repack
tff(fact_9834_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,I)),nth_VEBT_VEBTi(Tree_is,I))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList))),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% recomp
tff(fact_9835_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,Xa: vEBT_VEBTi,Xb3: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( Xaa = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xaa),Xa)),aa_o_assn(pure_assn,Xb3 = vEBT_vebt_mint(Xaa)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList))),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),aa_o_assn(pure_assn,Xb3 = vEBT_vebt_mint(Xaa)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,Xaa),list_u6098035379799741383_VEBTi(Tree_is,H,Xa)))) ) ) ).

% big_assn_simp'
tff(fact_9836_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
          & aa_nat_o(P,M3) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less_eq
tff(fact_9837_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
         => aa_nat_o(P,M3) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less_eq
tff(fact_9838_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9839_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = insert_nat(Na,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)) ).

% atLeast0_lessThan_Suc
tff(fact_9840_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9841_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),insert_nat(Na,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9842_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9843_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9844_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,Na)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9845_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xa: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ka,C),aa_nat_set_nat(set_or4665077453230672383an_nat(Xa),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9846_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(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_9847_Sum__Ico__nat,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cg),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = 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,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9848_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J3)) ) )
     => ( ! [I3: nat,J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J3)),aa_nat_nat(B,I3)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kb(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9849_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_9850_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_kc,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_9851_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastLessThan
tff(fact_9852_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_kd(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,minus_8373710615458151222nteger(U,L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_9853_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_9854_norm__assertion__simps_I6_J,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Xa),bot_bot_assn) = Xa ).

% norm_assertion_simps(6)
tff(fact_9855_norm__assertion__simps_I5_J,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),Xa) = Xa ).

% norm_assertion_simps(5)
tff(fact_9856_norm__assertion__simps_I9_J,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),Xa) = bot_bot_assn ).

% norm_assertion_simps(9)
tff(fact_9857_norm__assertion__simps_I10_J,axiom,
    ! [Xa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Xa),bot_bot_assn) = bot_bot_assn ).

% norm_assertion_simps(10)
tff(fact_9858_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),Xa)
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),vEBT_Leaf((A4),(B5))) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),vEBT_Node(Info,Deg2,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_9859_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),Xa)
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),vEBT_Leaf((A4),(B5))) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),vEBT_Node(Info,Deg2,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_9860_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),Xa)
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = zero_zero_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf((A4),(B5))) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,Deg2,TreeList2,Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList2))))) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Node(Uu2,Deg2,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_9861_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),Xa)
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = one_one_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),vEBT_Leaf((A4),(B5))) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),vEBT_Node(Info,Deg2,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_9862_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),Xa)
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((A4),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((A4),(B5))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy2,Uz2) )
                 => ( ( Y = 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(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_9863_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),Xa)
       => ( ! [A4: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B5)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((B5),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((A4),(B5))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy2,Uz2) )
                 => ( ( Y = 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(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_9864_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).

% take_bit_num_simps(1)
tff(fact_9865_take__bit__num__simps_I2_J,axiom,
    ! [Na: nat] : bit_take_bit_num(aa_nat_nat(suc,Na),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9866_take__bit__num__simps_I5_J,axiom,
    ! [R: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_9867_take__bit__num__simps_I3_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,Na),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ke,bit_take_bit_num(Na,M)) ).

% take_bit_num_simps(3)
tff(fact_9868_take__bit__num__simps_I4_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,Na),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Na,M))) ).

% take_bit_num_simps(4)
tff(fact_9869_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ke,bit_take_bit_num(aa_num_nat(pred_numeral,R),M)) ).

% take_bit_num_simps(6)
tff(fact_9870_take__bit__num__simps_I7_J,axiom,
    ! [R: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R),M))) ).

% take_bit_num_simps(7)
tff(fact_9871_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,Na: num] : bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = case_option_int_num(zero_zero_int,aTP_Lamm_kf(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),Na)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9872_and__minus__numerals_I7_J,axiom,
    ! [Na: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,Na))) ).

% and_minus_numerals(7)
tff(fact_9873_and__minus__numerals_I8_J,axiom,
    ! [Na: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,Na))) ).

% and_minus_numerals(8)
tff(fact_9874_and__minus__numerals_I4_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,Na))) ).

% and_minus_numerals(4)
tff(fact_9875_and__minus__numerals_I3_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,Na))) ).

% and_minus_numerals(3)
tff(fact_9876_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9877_and__not__num_Osimps_I2_J,axiom,
    ! [Na: num] : bit_and_not_num(one,aa_num_num(bit0,Na)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_9878_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit0,M),one) = some_num(aa_num_num(bit0,M)) ).

% and_not_num.simps(4)
tff(fact_9879_and__not__num_Osimps_I3_J,axiom,
    ! [Na: num] : bit_and_not_num(one,aa_num_num(bit1,Na)) = none_num ).

% and_not_num.simps(3)
tff(fact_9880_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit1,M),one) = some_num(aa_num_num(bit0,M)) ).

% and_not_num.simps(7)
tff(fact_9881_and__not__num__eq__Some__iff,axiom,
    ! [M: num,Na: num,Q2: num] :
      ( ( bit_and_not_num(M,Na) = some_num(Q2) )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9882_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,Na: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_kg,bit_and_not_num(M,Na)) ).

% and_not_num.simps(8)
tff(fact_9883_and__not__num__eq__None__iff,axiom,
    ! [M: num,Na: num] :
      ( ( bit_and_not_num(M,Na) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9884_int__numeral__and__not__num,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,Na)) ).

% int_numeral_and_not_num
tff(fact_9885_int__numeral__not__and__num,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Na,M)) ).

% int_numeral_not_and_num
tff(fact_9886_take__bit__num__def,axiom,
    ! [Na: nat,M: num] :
      bit_take_bit_num(Na,M) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,M))))) ).

% take_bit_num_def
tff(fact_9887_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_9888_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9889_numeral__num__of__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Na)) = Na ) ) ).

% numeral_num_of_nat
tff(fact_9890_num__of__nat__One,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),one_one_nat)
     => ( aa_nat_num(num_of_nat,Na) = one ) ) ).

% num_of_nat_One
tff(fact_9891_num__of__nat__double,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Na)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Na)) ) ) ).

% num_of_nat_double
tff(fact_9892_num__of__nat__plus__distrib,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,Na)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9893_num__of__nat_Osimps_I2_J,axiom,
    ! [Na: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na),aa_num_num(inc,aa_nat_num(num_of_nat,Na)),one) ).

% num_of_nat.simps(2)
tff(fact_9894_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se8568078237143864401it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9895_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se8568078237143864401it_int(Na,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_9896_drop__bit__minus__one,axiom,
    ! [Na: nat] : bit_se8568078237143864401it_int(Na,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9897_drop__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : bit_se8568078237143864401it_int(aa_nat_nat(suc,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = bit_se8568078237143864401it_int(Na,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9898_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(Na = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9899_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : 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)))) = 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_9900_drop__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : bit_se8568078237143864401it_int(aa_nat_nat(suc,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = bit_se8568078237143864401it_int(Na,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9901_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : 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)))) = 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_9902_drop__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,bit_se8568078237143864401it_int(Na,K)) ).

% drop_bit_nat_eq
tff(fact_9903_VEBT__internal_OperInsTrans_Ointros_I1_J,axiom,
    ! [T: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(T),T) ).

% VEBT_internal.perInsTrans.intros(1)
tff(fact_9904_drop__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se8568078237143864401it_int(Na,K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% drop_bit_int_def
tff(fact_9905_drop__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),M) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% drop_bit_nat_def
tff(fact_9906_VEBT__internal_OperInsTrans_Ointros_I2_J,axiom,
    ! [T: vEBT_VEBT,Sa: vEBT_VEBT,Xa: nat,U: vEBT_VEBT] :
      ( ( T = vEBT_vebt_insert(Sa,Xa) )
     => ( aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(T),U)
       => aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(Sa),U) ) ) ).

% VEBT_internal.perInsTrans.intros(2)
tff(fact_9907_VEBT__internal_OperInsTrans_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(A1),A22)
     => ( ( A22 != A1 )
       => ~ ! [T4: vEBT_VEBT] :
              ( ? [X: nat] : T4 = vEBT_vebt_insert(A1,X)
             => ~ aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(T4),A22) ) ) ) ).

% VEBT_internal.perInsTrans.cases
tff(fact_9908_VEBT__internal_OperInsTrans_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(A1),A22)
    <=> ( ? [T2: vEBT_VEBT] :
            ( ( A1 = T2 )
            & ( A22 = T2 ) )
        | ? [T2: vEBT_VEBT,S5: vEBT_VEBT,X4: nat,U2: vEBT_VEBT] :
            ( ( A1 = S5 )
            & ( A22 = U2 )
            & ( T2 = vEBT_vebt_insert(S5,X4) )
            & aa_VEBT_VEBT_o(vEBT_V6289311342943941716sTrans(T2),U2) ) ) ) ).

% VEBT_internal.perInsTrans.simps
tff(fact_9909_bin__rest__code,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,I),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = bit_se8568078237143864401it_int(one_one_nat,I) ).

% bin_rest_code
tff(fact_9910_shiftr__integer__conv__div__pow2,axiom,
    ! [Na: nat,Xa: code_integer] : bit_se3928097537394005634nteger(Na,Xa) = divide6298287555418463151nteger(Xa,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) ).

% shiftr_integer_conv_div_pow2
tff(fact_9911_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se545348938243370406it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9912_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se545348938243370406it_int(Na,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_9913_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),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))),Na) ).

% push_bit_of_Suc_0
tff(fact_9914_drop__bit__push__bit__int,axiom,
    ! [M: nat,Na: nat,K: int] : bit_se8568078237143864401it_int(M,bit_se545348938243370406it_int(Na,K)) = bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na),bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M),K)) ).

% drop_bit_push_bit_int
tff(fact_9915_flip__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9916_push__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,bit_se545348938243370406it_int(Na,K)) ).

% push_bit_nat_eq
tff(fact_9917_set__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9918_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(bit_se545348938243370406it_int(M,K)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9919_shiftl__integer__conv__mult__pow2,axiom,
    ! [Na: nat,Xa: code_integer] : bit_se7788150548672797655nteger(Na,Xa) = times_3573771949741848930nteger(Xa,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_9920_Bit__Operations_Oset__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se7879613467334960850it_int(Na,K) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),bit_se545348938243370406it_int(Na,one_one_int)) ).

% Bit_Operations.set_bit_int_def
tff(fact_9921_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q2)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9922_flip__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se2159334234014336723it_int(Na,K) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),bit_se545348938243370406it_int(Na,one_one_int)) ).

% flip_bit_int_def
tff(fact_9923_unset__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se4203085406695923979it_int(Na,K) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),bit_ri7919022796975470100ot_int(bit_se545348938243370406it_int(Na,one_one_int))) ).

% unset_bit_int_def
tff(fact_9924_push__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se545348938243370406it_int(Na,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% push_bit_int_def
tff(fact_9925_push__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% push_bit_nat_def
tff(fact_9926_push__bit__minus__one,axiom,
    ! [Na: nat] : bit_se545348938243370406it_int(Na,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% push_bit_minus_one
tff(fact_9927_Bit__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: $o] : bits_Bit_integer(aa_int_Code_integer(code_integer_of_int,Xa),(Xb3)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(Xb3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))) ).

% Bit_integer.abs_eq
tff(fact_9928_dup__1,axiom,
    code_dup(one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% dup_1
tff(fact_9929_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,I)
    <=> ( modulo364778990260209775nteger(I,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_nbe
tff(fact_9930_dup_Oabs__eq,axiom,
    ! [Xa: int] : code_dup(aa_int_Code_integer(code_integer_of_int,Xa)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xa)) ).

% dup.abs_eq
tff(fact_9931_bin__last__integer_Oabs__eq,axiom,
    ! [Xa: int] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,aa_int_Code_integer(code_integer_of_int,Xa))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ) ).

% bin_last_integer.abs_eq
tff(fact_9932_bin__rest__integer_Oabs__eq,axiom,
    ! [Xa: int] : bits_b2549910563261871055nteger(aa_int_Code_integer(code_integer_of_int,Xa)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% bin_rest_integer.abs_eq
tff(fact_9933_bin__rest__integer__code,axiom,
    ! [I: code_integer] : bits_b2549910563261871055nteger(I) = divide6298287555418463151nteger(I,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% bin_rest_integer_code
tff(fact_9934_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_9935_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,Na)),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(Na),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_9936_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_9937_concat__bit__of__zero__2,axiom,
    ! [Na: nat,K: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),zero_zero_int) = bit_se2923211474154528505it_int(Na,K) ).

% concat_bit_of_zero_2
tff(fact_9938_concat__bit__nonnegative__iff,axiom,
    ! [Na: 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(Na),K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_9939_concat__bit__negative__iff,axiom,
    ! [Na: 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(Na),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_9940_concat__bit__of__zero__1,axiom,
    ! [Na: nat,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),zero_zero_int),L) = bit_se545348938243370406it_int(Na,L) ).

% concat_bit_of_zero_1
tff(fact_9941_concat__bit__assoc,axiom,
    ! [Na: nat,K: int,M: nat,L: int,R: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),L),R)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),L)),R) ).

% concat_bit_assoc
tff(fact_9942_concat__bit__eq__iff,axiom,
    ! [Na: nat,K: int,L: int,R: int,Sa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),L) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),R),Sa) )
    <=> ( ( bit_se2923211474154528505it_int(Na,K) = bit_se2923211474154528505it_int(Na,R) )
        & ( L = Sa ) ) ) ).

% concat_bit_eq_iff
tff(fact_9943_concat__bit__take__bit__eq,axiom,
    ! [Na: nat,B: int] : aa_int_fun_int_int(bit_concat_bit(Na),bit_se2923211474154528505it_int(Na,B)) = aa_int_fun_int_int(bit_concat_bit(Na),B) ).

% concat_bit_take_bit_eq
tff(fact_9944_concat__bit__eq,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,bit_se2923211474154528505it_int(Na,K)),bit_se545348938243370406it_int(Na,L)) ).

% concat_bit_eq
tff(fact_9945_concat__bit__def,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,bit_se2923211474154528505it_int(Na,K)),bit_se545348938243370406it_int(Na,L)) ).

% concat_bit_def
tff(fact_9946_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),K),L)),Na)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
          & aa_nat_o(bit_se1146084159140164899it_int(K),Na) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9947_signed__take__bit__eq__concat__bit,axiom,
    ! [Na: nat,K: int] : bit_ri631733984087533419it_int(Na,K) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Na)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_9948_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) ).

% int_lsb_numeral(6)
tff(fact_9949_int__lsb__numeral_I3_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,one)) ).

% int_lsb_numeral(3)
tff(fact_9950_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))) ).

% int_lsb_numeral(7)
tff(fact_9951_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))) ).

% int_lsb_numeral(8)
tff(fact_9952_int__lsb__numeral_I5_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) ).

% int_lsb_numeral(5)
tff(fact_9953_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))) ).

% int_lsb_numeral(9)
tff(fact_9954_bin__last__conv__lsb,axiom,
    aTP_Lamm_kh = least_4859182151741483524sb_int ).

% bin_last_conv_lsb
tff(fact_9955_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(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_9956_vebt__assn__def,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia) = ex_assn_VEBT_VEBT(aa_VEB2970430004736285756T_assn(aa_set5977240410424667023T_assn(aTP_Lamm_ki(Na),Sa),Tia)) ).

% vebt_assn_def
tff(fact_9957_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9958_map__add__upt,axiom,
    ! [Na: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj,Na),upt(zero_zero_nat,M)) = upt(Na,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% map_add_upt
tff(fact_9959_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj,Ofs),upt(A,B)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ofs)) ).

% map_add_upt'
tff(fact_9960_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9961_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xa: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xa,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xa )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9962_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : map_nat_nat(aTP_Lamm_kk,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Na))) = upt(M,Na) ).

% map_decr_upt
tff(fact_9963_sdiv__int__numeral__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) ).

% sdiv_int_numeral_numeral
tff(fact_9964_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B) != zero_zero_int )
     => ( sgn_sgn_int(aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B)) = sgn_sgn_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_9965_signed__divide__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(K)),sgn_sgn_int(L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ).

% signed_divide_int_def
tff(fact_9966_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_9967_Cauchy__iff2,axiom,
    ! [X7: fun_nat_real] :
      ( topolo4055970368930404560y_real(X7)
    <=> ! [J2: nat] :
        ? [M8: nat] :
        ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M3)
         => ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N2)
             => 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(X7,M3)),aa_nat_real(X7,N2)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J2)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9968_range__mult,axiom,
    ! [A: real] :
      image_real_real(times_times_real(A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9969_infinite__UNIV__int,axiom,
    ~ aa_set_int_o(finite_finite_int,top_top_set_int) ).

% infinite_UNIV_int
tff(fact_9970_int__in__range__abs,axiom,
    ! [Na: nat] : aa_set_int_o(member_int(aa_nat_int(semiri1314217659103216013at_int,Na)),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_9971_range__mod,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kl,Na),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na) ) ) ).

% range_mod
tff(fact_9972_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_9973_root__def,axiom,
    ! [Na: nat,Xa: real] :
      aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = $ite(Na = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_km,Na)),Xa)) ).

% root_def
tff(fact_9974_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).

% int_of_nat_def
tff(fact_9975_smod__int__range,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,B))),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,B)),one_one_int))) ) ).

% smod_int_range
tff(fact_9976_smod__int__numeral__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) ).

% smod_int_numeral_numeral
tff(fact_9977_smod__int__compares_I8_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)) ) ) ).

% smod_int_compares(8)
tff(fact_9978_smod__int__compares_I7_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),zero_zero_int) ) ) ).

% smod_int_compares(7)
tff(fact_9979_smod__int__compares_I6_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)) ) ) ).

% smod_int_compares(6)
tff(fact_9980_smod__int__compares_I4_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),zero_zero_int) ) ) ).

% smod_int_compares(4)
tff(fact_9981_smod__int__compares_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)) ) ) ).

% smod_int_compares(2)
tff(fact_9982_smod__int__compares_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),B) ) ) ).

% smod_int_compares(1)
tff(fact_9983_smod__mod__positive,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% smod_mod_positive
tff(fact_9984_signed__modulo__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,K),L)),L)) ).

% signed_modulo_int_def
tff(fact_9985_smod__int__compares_I5_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),aa_int_int(uminus_uminus_int,B)) ) ) ).

% smod_int_compares(5)
tff(fact_9986_smod__int__compares_I3_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)) ) ) ).

% smod_int_compares(3)
tff(fact_9987_smod__int__alt__def,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(A)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% smod_int_alt_def
tff(fact_9988_uint32_Osize__eq__length,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) = type_l796852477590012082l_num1(type_N8448461349408098053l_num1) ).

% uint32.size_eq_length
tff(fact_9989_len__of__finite__3__def,axiom,
    ! [Xa: itself_finite_3] : type_l31302759751748493nite_3(Xa) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% len_of_finite_3_def
tff(fact_9990_len__of__finite__2__def,axiom,
    ! [Xa: itself_finite_2] : type_l31302759751748492nite_2(Xa) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% len_of_finite_2_def
tff(fact_9991_min__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)) ).

% min_Suc_Suc
tff(fact_9992_min__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Na) = zero_zero_nat ).

% min_0L
tff(fact_9993_min__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9994_min__minus,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus
tff(fact_9995_min__minus_H,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus'
tff(fact_9996_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),aa_nat_nat(suc,A)) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(2)
tff(fact_9997_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A)),B) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(1)
tff(fact_9998_min__pm,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ).

% min_pm
tff(fact_9999_min__pm1,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)) = A ).

% min_pm1
tff(fact_10000_rev__min__pm,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ).

% rev_min_pm
tff(fact_10001_rev__min__pm1,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A)) = A ).

% rev_min_pm1
tff(fact_10002_min__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% min_numeral_Suc
tff(fact_10003_min__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_10004_concat__bit__assoc__sym,axiom,
    ! [M: nat,Na: nat,K: int,L: int,R: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),L)),R) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),L),R)) ).

% concat_bit_assoc_sym
tff(fact_10005_min__diff,axiom,
    ! [M: nat,I: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)),I) ).

% min_diff
tff(fact_10006_nat__mult__min__left,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q2)) ).

% nat_mult_min_left
tff(fact_10007_nat__mult__min__right,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_min_right
tff(fact_10008_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_10009_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Na: nat,K: int,L: int] : bit_se2923211474154528505it_int(M,aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),L)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)),K),bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na),L)) ).

% take_bit_concat_bit_eq
tff(fact_10010_mod__mod__power,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na))) ).

% mod_mod_power
tff(fact_10011_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q2),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_10012_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,zero_z5237406670263579293d_enat),Q2) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_10013_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_10014_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
tff(fact_10015_int__set__bits__K__False,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_ao) = zero_zero_int ).

% int_set_bits_K_False
tff(fact_10016_int__set__bits__K__True,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_kn) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_set_bits_K_True
tff(fact_10017_bin__last__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_bi6516823479961619367ts_int(F))
      <=> aa_nat_o(F,zero_zero_nat) ) ) ).

% bin_last_set_bits
tff(fact_10018_wf__set__bits__int__Suc,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(aTP_Lamm_ko(F))
    <=> bit_wf_set_bits_int(F) ) ).

% wf_set_bits_int_Suc
tff(fact_10019_wf__set__bits__int__const,axiom,
    ! [B: $o] : bit_wf_set_bits_int(aTP_Lamm_kp((B))) ).

% wf_set_bits_int_const
tff(fact_10020_int__set__bits__unfold__BIT,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( bit_bi6516823479961619367ts_int(F) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(F,zero_zero_nat))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_10021_bin__rest__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,bit_bi6516823479961619367ts_int(F)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_10022_shiftl__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh3965577149348748681tl_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% shiftl_Suc_0
tff(fact_10023_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ~ aa_set_nat_o(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_10024_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_10025_less__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_10026_natural__of__integer_Orsp,axiom,
    bNF_re3715656647883201625at_nat(fequal_int,fequal_nat,nat2,nat2) ).

% natural_of_integer.rsp
tff(fact_10027_modulo__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),modulo_modulo_int,modulo_modulo_int) ).

% modulo_integer.rsp
tff(fact_10028_plus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),plus_plus_int,plus_plus_int) ).

% plus_integer.rsp
tff(fact_10029_dup_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_kq),aTP_Lamm_kq) ).

% dup.rsp
tff(fact_10030_times__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),times_times_int,times_times_int) ).

% times_integer.rsp
tff(fact_10031_divide__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),divide_divide_int,divide_divide_int) ).

% divide_integer.rsp
tff(fact_10032_num__of__integer_Orsp,axiom,
    bNF_re7626690874201225453um_num(fequal_int,fequal_num,comp_nat_num_int(num_of_nat,nat2),comp_nat_num_int(num_of_nat,nat2)) ).

% num_of_integer.rsp
tff(fact_10033_bin__rest__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_kr),aTP_Lamm_kr) ).

% bin_rest_integer.rsp
tff(fact_10034_bin__last__integer_Orsp,axiom,
    aa_fun_int_o_o(aa_fun1331825017273370463nt_o_o(bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),aTP_Lamm_kh),aTP_Lamm_kh) ).

% bin_last_integer.rsp
tff(fact_10035_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_ks,aTP_Lamm_ks) ).

% sub.rsp
tff(fact_10036_less__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_10037_integer__of__natural_Orsp,axiom,
    bNF_re6650684261131312217nt_int(fequal_nat,fequal_int,semiri1314217659103216013at_int,semiri1314217659103216013at_int) ).

% integer_of_natural.rsp
tff(fact_10038_inj__Suc,axiom,
    ! [N3: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N3) ).

% inj_Suc
tff(fact_10039_modulo__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),modulo_modulo_nat,modulo_modulo_nat) ).

% modulo_natural.rsp
tff(fact_10040_plus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),plus_plus_nat,plus_plus_nat) ).

% plus_natural.rsp
tff(fact_10041_times__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),times_times_nat,times_times_nat) ).

% times_natural.rsp
tff(fact_10042_divide__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),divide_divide_nat,divide_divide_nat) ).

% divide_natural.rsp
tff(fact_10043_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ka,K)),N3) ) ).

% inj_on_diff_nat
tff(fact_10044_Bit__integer_Orsp,axiom,
    bNF_re7138570853340013943_o_int(fequal_int,bNF_re1549289456480014903nt_int(fequal_o,fequal_int),aTP_Lamm_kt,aTP_Lamm_kt) ).

% Bit_integer.rsp
tff(fact_10045_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_km,Na)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_10046_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,M) = produc478579273971653890on_num(aTP_Lamm_kx,product_Pair_nat_num(Na,M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_10047_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,aa_num_num(bit1,M)) = case_nat_option_num(none_num,aTP_Lamm_ky(M),Na) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_10048_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,aa_num_num(bit0,M)) = case_nat_option_num(none_num,aTP_Lamm_kz(M),Na) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_10049_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_kn),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_10050_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_ao),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_10051_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),Na) ) ).

% less_eq_nat.simps(2)
tff(fact_10052_max__Suc2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_la,Na)),M) ).

% max_Suc2
tff(fact_10053_max__Suc1,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_lb,Na)),M) ).

% max_Suc1
tff(fact_10054_diff__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cg),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ).

% diff_Suc
tff(fact_10055_min__Suc1,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_lc,Na)),M) ).

% min_Suc1
tff(fact_10056_min__Suc2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_ld,Na)),M) ).

% min_Suc2
tff(fact_10057_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Na: nat] : bit_take_bit_num(Na,one) = case_nat_option_num(none_num,aTP_Lamm_le,Na) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_10058_DERIV__real__root__generic,axiom,
    ! [Na: nat,Xa: real,D3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( Xa != 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))),Na)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
             => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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))),Na)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
               => ( D3 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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))),Na)
               => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),D3,topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_10059_DERIV__pow,axiom,
    ! [Na: nat,Xa: real,Sa: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_lf,Na),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xa,Sa)) ).

% DERIV_pow
tff(fact_10060_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [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(Xa),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y2)),aa_real_real(F,Xa)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_10061_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [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(Xa),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xa)),aa_real_real(F,Y2)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_10062_DERIV__ln__divide,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(divide_divide_real(one_one_real),Xa),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_10063_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X,top_top_set_real))
       => ( aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(minus_minus_real(B),A)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_10064_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X,top_top_set_real))
       => ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_10065_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [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(Xa),Y2))),D)
             => ( aa_real_real(F,Xa) = aa_real_real(F,Y2) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_10066_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(minus_minus_real(Xa),H3))),aa_real_real(F,Xa)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_10067_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(minus_minus_real(Xa),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_10068_DERIV__ln,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xa),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_10069_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Xa),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_lg(F),aa_real_real(uminus_uminus_real,Y),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_10070_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_10071_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3))),aa_real_real(F,Xa)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_10072_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_10073_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_10074_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F2,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),aa_real_real(F2,Z3)) ) ) ) ) ).

% MVT2
tff(fact_10075_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_10076_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,Xa: real,Na: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_lh(G),Na),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)))),M),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_10077_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_10078_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3))),aa_real_real(F,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_10079_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(minus_minus_real(Xa),H3)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(minus_minus_real(Xa),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_10080_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(minus_minus_real(Xa),H3)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(minus_minus_real(Xa),H3))),aa_real_real(F,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_10081_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aTP_Lamm_li(R),aa_real_real(times_times_real(R),aa_real_real(powr_real(Z),aa_real_real(minus_minus_real(R),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_10082_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,Xa: real,R: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xa))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_lj(G),R),aa_real_real(times_times_real(aa_real_real(times_times_real(R),aa_real_real(powr_real(aa_real_real(G,Xa)),aa_real_real(minus_minus_real(R),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_10083_DERIV__log,axiom,
    ! [Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => 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)),Xa)),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_log
tff(fact_10084_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,Xa: real,F: fun_real_real,R: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xa))
       => ( has_fi5821293074295781190e_real(F,R,topolo2177554685111907308n_real(Xa,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_lk(G),F),aa_real_real(times_times_real(aa_real_real(powr_real(aa_real_real(G,Xa)),aa_real_real(F,Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(R),aa_real_real(ln_ln_real,aa_real_real(G,Xa)))),aa_real_real(divide_divide_real(aa_real_real(times_times_real(M),aa_real_real(F,Xa))),aa_real_real(G,Xa)))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_10085_DERIV__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_10086_DERIV__arctan,axiom,
    ! [Xa: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ).

% DERIV_arctan
tff(fact_10087_arsinh__real__has__field__derivative,axiom,
    ! [Xa: real,A2: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xa,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_10088_DERIV__real__sqrt__generic,axiom,
    ! [Xa: real,D3: real] :
      ( ( Xa != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( D3 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
           => ( D3 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D3,topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_10089_arcosh__real__has__field__derivative,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => 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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xa,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_10090_artanh__real__has__field__derivative,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xa,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_10091_DERIV__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_10092_DERIV__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_10093_DERIV__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_10094_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xa: real,Na: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M4: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X),topolo2177554685111907308n_real(X,top_top_set_real))
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ll(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_10095_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xa: real,Na: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M4: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X),topolo2177554685111907308n_real(X,top_top_set_real)) )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
          & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ll(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_10096_DERIV__odd__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( ( Xa != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_10097_Maclaurin__minus,axiom,
    ! [H: real,Na: 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),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M4: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_lm(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_10098_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: 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 )
       => ( ! [M4: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_lm(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_10099_Maclaurin,axiom,
    ! [H: real,Na: 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),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M4: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_lm(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_10100_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,Xa: 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),Na)
       => ( ( Xa != zero_zero_real )
         => ( ! [M4: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X),topolo2177554685111907308n_real(X,top_top_set_real))
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
                & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ll(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_10101_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,Xa: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M4: nat,T4: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ll(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_10102_Taylor__down,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C)
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ln(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(A),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_10103_Taylor__up,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),B)
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ln(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(B),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_10104_Taylor,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xa)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),B)
                 => ( ( Xa != C )
                   => ? [T4: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xa) ) )
                        & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_lo(Diff),C),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xa),C)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_10105_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M4: nat,T4: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
     => ( ( Na = aa_nat_nat(suc,K) )
       => ! [M2: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_lq(Na),Diff),B2),M2),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lr(Diff),M2),T5),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2))))),aa_real_real(times_times_real(B2),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,T5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T5,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_10106_DERIV__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_ls,suminf_real(aa_real_fun_nat_real(aTP_Lamm_lt,Xa)),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_10107_DERIV__even__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_10108_DERIV__power__series_H,axiom,
    ! [R2: real,F: fun_nat_real,X0: real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_lu(F),X)) )
     => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
         => has_fi5821293074295781190e_real(aTP_Lamm_lw(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_lu(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_10109_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_10110_tanh__real__bounds,axiom,
    ! [Xa: real] : aa_set_real_o(member_real(aa_real_real(tanh_real,Xa)),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) ).

% tanh_real_bounds
tff(fact_10111_DERIV__isconst3,axiom,
    ! [A: real,B: real,Xa: real,Y: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_set_real_o(member_real(Xa),set_or1633881224788618240n_real(A,B))
       => ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(A,B))
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X,top_top_set_real)) )
           => ( aa_real_real(F,Xa) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_10112_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F2: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L4: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_lx(F),N),aa_nat_real(aa_real_fun_nat_real(F2,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X)) )
       => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F2,X0))
           => ( summable_real(L4)
             => ( ! [N: nat,X: real,Y2: real] :
                    ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
                   => ( aa_set_real_o(member_real(Y2),set_or1633881224788618240n_real(A,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(F,X),N)),aa_nat_real(aa_real_fun_nat_real(F,Y2),N)))),aa_real_real(times_times_real(aa_nat_real(L4,N)),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y2)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_ly(F),suminf_real(aa_real_fun_nat_real(F2,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_10113_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(C),X5))),R4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X5)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_10114_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(C),X5))),R4) )
               => ( aa_real_real(F,X5) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_10115_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(C),X5))),R4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X5)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_10116_isCont__arctan,axiom,
    ! [Xa: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),arctan) ).

% isCont_arctan
tff(fact_10117_isCont__real__sqrt,axiom,
    ! [Xa: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_10118_isCont__real__root,axiom,
    ! [Xa: real,Na: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),aa_nat_fun_real_real(root,Na)) ).

% isCont_real_root
tff(fact_10119_isCont__arsinh,axiom,
    ! [Xa: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),arsinh_real) ).

% isCont_arsinh
tff(fact_10120_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_10121_isCont__inverse__function2,axiom,
    ! [A: real,Xa: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),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,Xa),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_10122_isCont__ln,axiom,
    ! [Xa: real] :
      ( ( Xa != zero_zero_real )
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),ln_ln_real) ) ).

% isCont_ln
tff(fact_10123_isCont__arcosh,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_10124_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_lz,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_10125_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D3: real,G: fun_real_real,Xa: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D3,topolo2177554685111907308n_real(aa_real_real(G,Xa),top_top_set_real))
     => ( ( D3 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
           => ( ! [Y2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),B)
                   => ( aa_real_real(F,aa_real_real(G,Y2)) = Y2 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D3),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_10126_isCont__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_10127_isCont__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_10128_LIM__less__bound,axiom,
    ! [B: real,Xa: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Xa)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(B,Xa))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xa)) ) ) ) ).

% LIM_less_bound
tff(fact_10129_isCont__artanh,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_10130_isCont__inverse__function,axiom,
    ! [D: real,Xa: 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),Xa))),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),Xa))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xa),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_10131_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),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,A),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,A),Z3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
             => ? [C4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C4),B)
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C4)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F2,C4)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_10132_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N9: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_ma(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ma(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N9)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_ma(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N9))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_10133_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N9: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_ma(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ma(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N9))),groups6591440286371151544t_real(aTP_Lamm_ma(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N9)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_10134_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_10135_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_cw,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_10136_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_10137_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_mb,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_10138_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_fg(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L2: real] :
                ( ! [N9: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N9)),L2)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L2),at_top_nat)
                & ! [N9: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),aa_nat_real(G,N9))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_10139_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun_nat_real] :
      ( ! [R4: real] :
        ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R4),aa_nat_real(X7,N)) )
     => filterlim_nat_real(aTP_Lamm_mc(X7),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_10140_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_md,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_10141_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_me,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_10142_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_mf,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_10143_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mg,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_10144_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)
             => ? [N9: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N9)),E2)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_10145_LIMSEQ__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_10146_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mh(Xa),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_10147_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_10148_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_10149_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mi,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_10150_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mj,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_10151_tendsto__exp__limit__sequentially,axiom,
    ! [Xa: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mk,Xa),topolo2815343760600316023s_real(aa_real_real(exp_real,Xa)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_10152_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ml,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_10153_summable__Leibniz_I1_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => summable_real(aTP_Lamm_ma(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_10154_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,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(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => summable_real(aTP_Lamm_ma(A)) ) ) ) ).

% summable
tff(fact_10155_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mm(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_mn(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_10156_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_mo(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_mn(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_10157_summable__Leibniz_I4_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_mp(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ma(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_10158_zeroseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_dp,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_10159_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(A,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(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_ma(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),suminf_real(aTP_Lamm_ma(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_10160_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,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(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_mp(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ma(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_10161_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L2: real] :
              ( ! [N9: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_ma(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N9)))),L2)
              & filterlim_nat_real(aTP_Lamm_mp(A),topolo2815343760600316023s_real(L2),at_top_nat)
              & ! [N9: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),groups6591440286371151544t_real(aTP_Lamm_ma(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N9)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_mq(A),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_10162_summable__Leibniz_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_mq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ma(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_10163_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(A,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(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_ma(A))),groups6591440286371151544t_real(aTP_Lamm_ma(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_10164_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,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(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_mq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ma(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_10165_exp__at__top,axiom,
    filterlim_real_real(exp_real,at_top_real,at_top_real) ).

% exp_at_top
tff(fact_10166_ln__at__top,axiom,
    filterlim_real_real(ln_ln_real,at_top_real,at_top_real) ).

% ln_at_top
tff(fact_10167_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_10168_sinh__real__at__top,axiom,
    filterlim_real_real(sinh_real,at_top_real,at_top_real) ).

% sinh_real_at_top
tff(fact_10169_arsinh__real__at__top,axiom,
    filterlim_real_real(arsinh_real,at_top_real,at_top_real) ).

% arsinh_real_at_top
tff(fact_10170_filterlim__nat__sequentially,axiom,
    filterlim_int_nat(nat2,at_top_nat,at_top_int) ).

% filterlim_nat_sequentially
tff(fact_10171_arcosh__real__at__top,axiom,
    filterlim_real_real(arcosh_real,at_top_real,at_top_real) ).

% arcosh_real_at_top
tff(fact_10172_filterlim__int__sequentially,axiom,
    filterlim_nat_int(semiri1314217659103216013at_int,at_top_int,at_top_nat) ).

% filterlim_int_sequentially
tff(fact_10173_cosh__real__at__top,axiom,
    filterlim_real_real(cosh_real,at_top_real,at_top_real) ).

% cosh_real_at_top
tff(fact_10174_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C5: real] :
        ! [X4: real] : aa_real_real(F,X4) = aa_real_real(times_times_real(X4),C5) ) ).

% real_bounded_linear
tff(fact_10175_filterlim__real__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_top_real,at_top_nat) ).

% filterlim_real_sequentially
tff(fact_10176_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_10177_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_10178_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_mr,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_10179_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_ms,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_10180_tendsto__exp__limit__at__top,axiom,
    ! [Xa: real] : filterlim_real_real(aTP_Lamm_mt(Xa),topolo2815343760600316023s_real(aa_real_real(exp_real,Xa)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10181_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_10182_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_10183_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),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_10184_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( 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))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(Na),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_10185_tendsto__exp__limit__at__right,axiom,
    ! [Xa: real] : filterlim_real_real(aTP_Lamm_mv(Xa),topolo2815343760600316023s_real(aa_real_real(exp_real,Xa)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_10186_arsinh__real__at__bot,axiom,
    filterlim_real_real(arsinh_real,at_bot_real,at_bot_real) ).

% arsinh_real_at_bot
tff(fact_10187_sinh__real__at__bot,axiom,
    filterlim_real_real(sinh_real,at_bot_real,at_bot_real) ).

% sinh_real_at_bot
tff(fact_10188_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_10189_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_10190_cosh__real__at__bot,axiom,
    filterlim_real_real(cosh_real,at_top_real,at_bot_real) ).

% cosh_real_at_bot
tff(fact_10191_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_10192_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_10193_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_10194_exp__at__bot,axiom,
    filterlim_real_real(exp_real,topolo2815343760600316023s_real(zero_zero_real),at_bot_real) ).

% exp_at_bot
tff(fact_10195_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_10196_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_10197_log__inj,axiom,
    ! [B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => aa_set_real_o(inj_on_real_real(log(B)),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_10198_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_10199_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_10200_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_10201_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) )
     => ( 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_10202_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( 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))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(Na),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_10203_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_10204_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,Xa: real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
     => ( eventually_real(aTP_Lamm_mw(G2),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mx(F),F2),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mx(G),G2),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_my(G2),F2),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_my(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_10205_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,Xa: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_mw(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mx(F),F2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mx(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_my(G2),F2),topolo2815343760600316023s_real(Xa),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_my(G),F),topolo2815343760600316023s_real(Xa),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_10206_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ge,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_10207_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_hn,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_10208_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_jg,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_10209_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ht,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_10210_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_lz,Uu) = aa_real_real(divide_divide_real(aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10211_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_mr,Uu) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_10212_ATP_Olambda__7,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_kr,Uu) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Uu),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% ATP.lambda_7
tff(fact_10213_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_mb,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_8
tff(fact_10214_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_kk,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_9
tff(fact_10215_ATP_Olambda__10,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_kq,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uu) ).

% ATP.lambda_10
tff(fact_10216_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_md,Uu) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_11
tff(fact_10217_ATP_Olambda__12,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_kh,Uu)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uu) ) ).

% ATP.lambda_12
tff(fact_10218_ATP_Olambda__13,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ls,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ef,Uu)) ).

% ATP.lambda_13
tff(fact_10219_ATP_Olambda__14,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_me,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_14
tff(fact_10220_ATP_Olambda__15,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_kg,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_15
tff(fact_10221_ATP_Olambda__16,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_ke,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_16
tff(fact_10222_ATP_Olambda__17,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_jm,Uu) = product_Pair_int_int(aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_17
tff(fact_10223_ATP_Olambda__18,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_jp,Uu) = product_Pair_int_int(aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_18
tff(fact_10224_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_iw,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_19
tff(fact_10225_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_le,Uu) = some_num(one) ).

% ATP.lambda_20
tff(fact_10226_ATP_Olambda__21,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_kw(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_ku(Uua),aTP_Lamm_kv(Uua),Uu) ).

% ATP.lambda_21
tff(fact_10227_ATP_Olambda__22,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_ho(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_10228_ATP_Olambda__23,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jl,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_23
tff(fact_10229_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_kx,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_kw(Uua),Uu) ).

% ATP.lambda_24
tff(fact_10230_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ku(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ke,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_25
tff(fact_10231_ATP_Olambda__26,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_kz(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ke,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_26
tff(fact_10232_ATP_Olambda__27,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fv,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_10233_ATP_Olambda__28,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hx,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_10234_ATP_Olambda__29,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hw,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_10235_ATP_Olambda__30,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_jh,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_10236_ATP_Olambda__31,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_jd,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_10237_ATP_Olambda__32,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_jk(Uu),Uua)
    <=> ( ( sgn_sgn_complex(Uu) = 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_32
tff(fact_10238_ATP_Olambda__33,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_ji(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_33
tff(fact_10239_ATP_Olambda__34,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_jj(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_34
tff(fact_10240_ATP_Olambda__35,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ef,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_35
tff(fact_10241_ATP_Olambda__36,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lt,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_36
tff(fact_10242_ATP_Olambda__37,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ma(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_37
tff(fact_10243_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_il,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_38
tff(fact_10244_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hz,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_39
tff(fact_10245_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ms,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_40
tff(fact_10246_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ie,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_41
tff(fact_10247_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_id,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_42
tff(fact_10248_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_gz(Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_43
tff(fact_10249_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_gy(Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_44
tff(fact_10250_ATP_Olambda__45,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mk,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_45
tff(fact_10251_ATP_Olambda__46,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_mv(Uu),Uua) = aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_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_46
tff(fact_10252_ATP_Olambda__47,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_mt(Uu),Uua) = aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(Uu),Uua))),Uua) ).

% ATP.lambda_47
tff(fact_10253_ATP_Olambda__48,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dp,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_48
tff(fact_10254_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_jf,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_49
tff(fact_10255_ATP_Olambda__50,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_fn(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_50
tff(fact_10256_ATP_Olambda__51,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fo(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_51
tff(fact_10257_ATP_Olambda__52,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_fp(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_52
tff(fact_10258_ATP_Olambda__53,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mq(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_ma(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_53
tff(fact_10259_ATP_Olambda__54,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mp(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_ma(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_54
tff(fact_10260_ATP_Olambda__55,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_kf(Uu),Uua) = bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_55
tff(fact_10261_ATP_Olambda__56,axiom,
    ! [Uu: int,Uua: $o] : aa_o_int(aa_int_fun_o_int(aTP_Lamm_kt,Uu),(Uua)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uu)) ).

% ATP.lambda_56
tff(fact_10262_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_km,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_57
tff(fact_10263_ATP_Olambda__58,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hy,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_58
tff(fact_10264_ATP_Olambda__59,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hu,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_59
tff(fact_10265_ATP_Olambda__60,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_ks,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu)),aa_num_int(numeral_numeral_int,Uua)) ).

% ATP.lambda_60
tff(fact_10266_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_je,Uu),Uua)
    <=> ( aa_real_real(exp_real,Uua) = Uu ) ) ).

% ATP.lambda_61
tff(fact_10267_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_im,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_62
tff(fact_10268_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ml,Uu),Uua) = aa_real_real(times_times_real(Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_63
tff(fact_10269_ATP_Olambda__64,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jc,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_64
tff(fact_10270_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mj,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_65
tff(fact_10271_ATP_Olambda__66,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mg,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_66
tff(fact_10272_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_he,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_67
tff(fact_10273_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_68
tff(fact_10274_ATP_Olambda__69,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(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_69
tff(fact_10275_ATP_Olambda__70,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_cu(Uu),Uua) = aa_real_real(times_times_real(Uua),Uu) ).

% ATP.lambda_70
tff(fact_10276_ATP_Olambda__71,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_cv,Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Uua),Uu) ).

% ATP.lambda_71
tff(fact_10277_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cw,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_72
tff(fact_10278_ATP_Olambda__73,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_cx,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).

% ATP.lambda_73
tff(fact_10279_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ka,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_74
tff(fact_10280_ATP_Olambda__75,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_lf,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_75
tff(fact_10281_ATP_Olambda__76,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_kd(Uu),Uua) = plus_p5714425477246183910nteger(Uua,Uu) ).

% ATP.lambda_76
tff(fact_10282_ATP_Olambda__77,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_77
tff(fact_10283_ATP_Olambda__78,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_kc,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_78
tff(fact_10284_ATP_Olambda__79,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_li(Uu),Uua) = aa_real_real(powr_real(Uua),Uu) ).

% ATP.lambda_79
tff(fact_10285_ATP_Olambda__80,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_fy(Uu),Uua)
    <=> aa_complex_o(dvd_dvd_complex(Uua),Uu) ) ).

% ATP.lambda_80
tff(fact_10286_ATP_Olambda__81,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ga,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_81
tff(fact_10287_ATP_Olambda__82,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fz,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_82
tff(fact_10288_ATP_Olambda__83,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ib,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_83
tff(fact_10289_ATP_Olambda__84,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mf,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_84
tff(fact_10290_ATP_Olambda__85,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_aa(Uu),Uua)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uua),Uu) ) ).

% ATP.lambda_85
tff(fact_10291_ATP_Olambda__86,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_86
tff(fact_10292_ATP_Olambda__87,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_87
tff(fact_10293_ATP_Olambda__88,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_88
tff(fact_10294_ATP_Olambda__89,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_89
tff(fact_10295_ATP_Olambda__90,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ac(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_90
tff(fact_10296_ATP_Olambda__91,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_af(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_91
tff(fact_10297_ATP_Olambda__92,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ah(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_92
tff(fact_10298_ATP_Olambda__93,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ag(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_93
tff(fact_10299_ATP_Olambda__94,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ai,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_94
tff(fact_10300_ATP_Olambda__95,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aj,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_95
tff(fact_10301_ATP_Olambda__96,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ak,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_96
tff(fact_10302_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hs(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_97
tff(fact_10303_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hr(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_98
tff(fact_10304_ATP_Olambda__99,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_lg(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_99
tff(fact_10305_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_fi(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_100
tff(fact_10306_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fh(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_101
tff(fact_10307_ATP_Olambda__102,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ko(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_102
tff(fact_10308_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_kv(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_103
tff(fact_10309_ATP_Olambda__104,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_ky(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_104
tff(fact_10310_ATP_Olambda__105,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_lw(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_lv(Uu),Uua)) ).

% ATP.lambda_105
tff(fact_10311_ATP_Olambda__106,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ix(Uu),Uua) = 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_106
tff(fact_10312_ATP_Olambda__107,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_mw(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_107
tff(fact_10313_ATP_Olambda__108,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ja,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_108
tff(fact_10314_ATP_Olambda__109,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mi,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_109
tff(fact_10315_ATP_Olambda__110,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lc,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_110
tff(fact_10316_ATP_Olambda__111,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ld,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_111
tff(fact_10317_ATP_Olambda__112,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lb,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_112
tff(fact_10318_ATP_Olambda__113,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_la,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_113
tff(fact_10319_ATP_Olambda__114,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_dx(Uu),Uua)
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uua),Uu) ) ).

% ATP.lambda_114
tff(fact_10320_ATP_Olambda__115,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_eb(Uu),Uua)
    <=> ~ member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_115
tff(fact_10321_ATP_Olambda__116,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ea(Uu),Uua)
    <=> ~ aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_116
tff(fact_10322_ATP_Olambda__117,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_dw(Uu),Uua)
    <=> ~ aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_117
tff(fact_10323_ATP_Olambda__118,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_dy(Uu),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_118
tff(fact_10324_ATP_Olambda__119,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_dz(Uu),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_119
tff(fact_10325_ATP_Olambda__120,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_fx(Uu),Uua) = real_V1022390504157884413omplex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_120
tff(fact_10326_ATP_Olambda__121,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fw(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_121
tff(fact_10327_ATP_Olambda__122,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mc(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_122
tff(fact_10328_ATP_Olambda__123,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_eq(Uu),Uua) = real_V4546457046886955230omplex(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_123
tff(fact_10329_ATP_Olambda__124,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fm(Uu),Uua) = aa_real_real(real_V1803761363581548252l_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_124
tff(fact_10330_ATP_Olambda__125,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_fk(Uu),Uua) = uminus1482373934393186551omplex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_125
tff(fact_10331_ATP_Olambda__126,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fj(Uu),Uua) = aa_real_real(uminus_uminus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_126
tff(fact_10332_ATP_Olambda__127,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fl(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_127
tff(fact_10333_ATP_Olambda__128,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mo(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_128
tff(fact_10334_ATP_Olambda__129,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_ly(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_129
tff(fact_10335_ATP_Olambda__130,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_dv(Uu),Uua)
    <=> ~ aa_Pro4592068669816069595_int_o(Uu,Uua) ) ).

% ATP.lambda_130
tff(fact_10336_ATP_Olambda__131,axiom,
    ! [Uu: fun_complex_o,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_du(Uu),Uua)
    <=> ~ aa_complex_o(Uu,Uua) ) ).

% ATP.lambda_131
tff(fact_10337_ATP_Olambda__132,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ds(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_132
tff(fact_10338_ATP_Olambda__133,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_dt(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_133
tff(fact_10339_ATP_Olambda__134,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hp(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_134
tff(fact_10340_ATP_Olambda__135,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hq(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_135
tff(fact_10341_ATP_Olambda__136,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_iv(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(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(times_3573771949741848930nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),Uua),Uub)) ).

% ATP.lambda_136
tff(fact_10342_ATP_Olambda__137,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_it(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_137
tff(fact_10343_ATP_Olambda__138,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_is(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(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(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_138
tff(fact_10344_ATP_Olambda__139,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gp(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_139
tff(fact_10345_ATP_Olambda__140,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gq(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_140
tff(fact_10346_ATP_Olambda__141,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gr(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_141
tff(fact_10347_ATP_Olambda__142,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_gs(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_142
tff(fact_10348_ATP_Olambda__143,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_er(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_143
tff(fact_10349_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_es(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_144
tff(fact_10350_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_et(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_145
tff(fact_10351_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_eu(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_146
tff(fact_10352_ATP_Olambda__147,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gl(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_147
tff(fact_10353_ATP_Olambda__148,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gm(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_148
tff(fact_10354_ATP_Olambda__149,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gn(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_149
tff(fact_10355_ATP_Olambda__150,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_go(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_150
tff(fact_10356_ATP_Olambda__151,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_mx(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_151
tff(fact_10357_ATP_Olambda__152,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_lx(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_152
tff(fact_10358_ATP_Olambda__153,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jz(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_jy(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_153
tff(fact_10359_ATP_Olambda__154,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jx(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_jw(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_154
tff(fact_10360_ATP_Olambda__155,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jv(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ju(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_155
tff(fact_10361_ATP_Olambda__156,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jt(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_js(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_156
tff(fact_10362_ATP_Olambda__157,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_jr(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_jq(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_157
tff(fact_10363_ATP_Olambda__158,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_jo(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_jn(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_158
tff(fact_10364_ATP_Olambda__159,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ll(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_159
tff(fact_10365_ATP_Olambda__160,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_lm(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_160
tff(fact_10366_ATP_Olambda__161,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lu(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_161
tff(fact_10367_ATP_Olambda__162,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hv(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_162
tff(fact_10368_ATP_Olambda__163,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_if(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_163
tff(fact_10369_ATP_Olambda__164,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_bh(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_164
tff(fact_10370_ATP_Olambda__165,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_bk(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        | member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_165
tff(fact_10371_ATP_Olambda__166,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_bj(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_166
tff(fact_10372_ATP_Olambda__167,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_bg(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_167
tff(fact_10373_ATP_Olambda__168,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_bl(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_168
tff(fact_10374_ATP_Olambda__169,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_bi(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_169
tff(fact_10375_ATP_Olambda__170,axiom,
    ! [Uu: vEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_ax(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_170
tff(fact_10376_ATP_Olambda__171,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_bb(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_171
tff(fact_10377_ATP_Olambda__172,axiom,
    ! [Uu: complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_ba(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_172
tff(fact_10378_ATP_Olambda__173,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_aw(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_173
tff(fact_10379_ATP_Olambda__174,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ay(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_174
tff(fact_10380_ATP_Olambda__175,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_az(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_175
tff(fact_10381_ATP_Olambda__176,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_iy(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_176
tff(fact_10382_ATP_Olambda__177,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hg(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_177
tff(fact_10383_ATP_Olambda__178,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_iz(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_178
tff(fact_10384_ATP_Olambda__179,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hj(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_179
tff(fact_10385_ATP_Olambda__180,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hk(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_180
tff(fact_10386_ATP_Olambda__181,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hh(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_181
tff(fact_10387_ATP_Olambda__182,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ed(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_182
tff(fact_10388_ATP_Olambda__183,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ec(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_183
tff(fact_10389_ATP_Olambda__184,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_dd(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_184
tff(fact_10390_ATP_Olambda__185,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_dg(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_185
tff(fact_10391_ATP_Olambda__186,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_df(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_186
tff(fact_10392_ATP_Olambda__187,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_dc(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_187
tff(fact_10393_ATP_Olambda__188,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_dh(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_188
tff(fact_10394_ATP_Olambda__189,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_de(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_189
tff(fact_10395_ATP_Olambda__190,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: set_complex] :
      ( aa_set_complex_o(aa_set8643480048548153067plex_o(aTP_Lamm_gu(Uu),Uua),Uub)
    <=> ( minus_811609699411566653omplex(Uub,Uu) = minus_811609699411566653omplex(Uua,Uu) ) ) ).

% ATP.lambda_190
tff(fact_10396_ATP_Olambda__191,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_gv(Uu),Uua),Uub)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Uub),Uu) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Uua),Uu) ) ) ).

% ATP.lambda_191
tff(fact_10397_ATP_Olambda__192,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_gt(Uu),Uua),Uub)
    <=> ( minus_minus_set_int(Uub,Uu) = minus_minus_set_int(Uua,Uu) ) ) ).

% ATP.lambda_192
tff(fact_10398_ATP_Olambda__193,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_cj(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_193
tff(fact_10399_ATP_Olambda__194,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_cm(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_194
tff(fact_10400_ATP_Olambda__195,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_cl(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_195
tff(fact_10401_ATP_Olambda__196,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_ci(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_196
tff(fact_10402_ATP_Olambda__197,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cn(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_197
tff(fact_10403_ATP_Olambda__198,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_198
tff(fact_10404_ATP_Olambda__199,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_bm(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_199
tff(fact_10405_ATP_Olambda__200,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_bo(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_200
tff(fact_10406_ATP_Olambda__201,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_bn(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_201
tff(fact_10407_ATP_Olambda__202,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_202
tff(fact_10408_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_203
tff(fact_10409_ATP_Olambda__204,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_br(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_204
tff(fact_10410_ATP_Olambda__205,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_bs(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_205
tff(fact_10411_ATP_Olambda__206,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_bu(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_206
tff(fact_10412_ATP_Olambda__207,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_207
tff(fact_10413_ATP_Olambda__208,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bv(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_208
tff(fact_10414_ATP_Olambda__209,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bw(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_209
tff(fact_10415_ATP_Olambda__210,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_210
tff(fact_10416_ATP_Olambda__211,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : aa_o_assn(aa_VEB6687300313434771872o_assn(aTP_Lamm_dj(Uu),Uua),(Uub)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,(Uub) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Uu))) ).

% ATP.lambda_211
tff(fact_10417_ATP_Olambda__212,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_ee(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uub = vEBT_vebt_mint(Uu))) ).

% ATP.lambda_212
tff(fact_10418_ATP_Olambda__213,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_ia(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uub = vEBT_vebt_maxt(Uu))) ).

% ATP.lambda_213
tff(fact_10419_ATP_Olambda__214,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_cp(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_214
tff(fact_10420_ATP_Olambda__215,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_cs(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & ~ member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_215
tff(fact_10421_ATP_Olambda__216,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_cr(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ~ aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_216
tff(fact_10422_ATP_Olambda__217,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ~ aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_217
tff(fact_10423_ATP_Olambda__218,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ct(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_218
tff(fact_10424_ATP_Olambda__219,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_cq(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_219
tff(fact_10425_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ic(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_220
tff(fact_10426_ATP_Olambda__221,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_hi(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_221
tff(fact_10427_ATP_Olambda__222,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_jb(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_222
tff(fact_10428_ATP_Olambda__223,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fq(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_223
tff(fact_10429_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fs(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_224
tff(fact_10430_ATP_Olambda__225,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_mn(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_225
tff(fact_10431_ATP_Olambda__226,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lv(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_226
tff(fact_10432_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fr(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_227
tff(fact_10433_ATP_Olambda__228,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ek(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_228
tff(fact_10434_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ii(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_229
tff(fact_10435_ATP_Olambda__230,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gw(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_230
tff(fact_10436_ATP_Olambda__231,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_my(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_231
tff(fact_10437_ATP_Olambda__232,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kb(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_232
tff(fact_10438_ATP_Olambda__233,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ff(Uu),Uua),Uub) = minus_minus_complex(aa_nat_complex(Uu,Uub),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_233
tff(fact_10439_ATP_Olambda__234,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fg(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_234
tff(fact_10440_ATP_Olambda__235,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ex(Uu),Uua),Uub) = plus_plus_complex(aa_nat_complex(Uu,Uub),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_235
tff(fact_10441_ATP_Olambda__236,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ey(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_236
tff(fact_10442_ATP_Olambda__237,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ez(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_237
tff(fact_10443_ATP_Olambda__238,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_fa(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_238
tff(fact_10444_ATP_Olambda__239,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_lk(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_239
tff(fact_10445_ATP_Olambda__240,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_dn(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
       => aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_240
tff(fact_10446_ATP_Olambda__241,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_dm(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
       => aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_241
tff(fact_10447_ATP_Olambda__242,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_do(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_242
tff(fact_10448_ATP_Olambda__243,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_dl(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_243
tff(fact_10449_ATP_Olambda__244,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_be(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        | aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_244
tff(fact_10450_ATP_Olambda__245,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_bd(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_245
tff(fact_10451_ATP_Olambda__246,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bf(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_246
tff(fact_10452_ATP_Olambda__247,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bc(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_247
tff(fact_10453_ATP_Olambda__248,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_da(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_248
tff(fact_10454_ATP_Olambda__249,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_cz(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_249
tff(fact_10455_ATP_Olambda__250,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_db(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_250
tff(fact_10456_ATP_Olambda__251,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cy(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_251
tff(fact_10457_ATP_Olambda__252,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gx(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_252
tff(fact_10458_ATP_Olambda__253,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_eg(Uu),Uua),Uub) = divide1717551699836669952omplex(aa_nat_complex(Uu,Uub),Uua) ).

% ATP.lambda_253
tff(fact_10459_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_eh(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_254
tff(fact_10460_ATP_Olambda__255,axiom,
    ! [Uu: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_hc(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(Uu,Uub)),Uua) ).

% ATP.lambda_255
tff(fact_10461_ATP_Olambda__256,axiom,
    ! [Uu: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_hb(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_option_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_256
tff(fact_10462_ATP_Olambda__257,axiom,
    ! [Uu: fun_o_assn,Uua: assn,Uub: $o] : aa_o_assn(aa_assn_fun_o_assn(aTP_Lamm_ha(Uu),Uua),(Uub)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(Uu,(Uub))),Uua) ).

% ATP.lambda_257
tff(fact_10463_ATP_Olambda__258,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fb(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_258
tff(fact_10464_ATP_Olambda__259,axiom,
    ! [Uu: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_hd(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_259
tff(fact_10465_ATP_Olambda__260,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fc(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_260
tff(fact_10466_ATP_Olambda__261,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_lh(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_261
tff(fact_10467_ATP_Olambda__262,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_262
tff(fact_10468_ATP_Olambda__263,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_lj(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_263
tff(fact_10469_ATP_Olambda__264,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_264
tff(fact_10470_ATP_Olambda__265,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_av(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_265
tff(fact_10471_ATP_Olambda__266,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_au(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_266
tff(fact_10472_ATP_Olambda__267,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_267
tff(fact_10473_ATP_Olambda__268,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_268
tff(fact_10474_ATP_Olambda__269,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_269
tff(fact_10475_ATP_Olambda__270,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_gd(Uu),Uua),Uub) = aa_real_real(times_times_real(real_V1022390504157884413omplex(aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_270
tff(fact_10476_ATP_Olambda__271,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mh(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_271
tff(fact_10477_ATP_Olambda__272,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hl(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_272
tff(fact_10478_ATP_Olambda__273,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ei(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_273
tff(fact_10479_ATP_Olambda__274,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ej(Uu),Uua),Uub) = aa_real_real(times_times_real(Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_274
tff(fact_10480_ATP_Olambda__275,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fd(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uua),aa_nat_complex(Uu,Uub)) ).

% ATP.lambda_275
tff(fact_10481_ATP_Olambda__276,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fe(Uu),Uua),Uub) = aa_real_real(times_times_real(Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_276
tff(fact_10482_ATP_Olambda__277,axiom,
    ! [Uu: assn,Uua: vEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(aTP_Lamm_gj(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_277
tff(fact_10483_ATP_Olambda__278,axiom,
    ! [Uu: assn,Uua: option_nat,Uub: option_nat] : aa_option_nat_assn(aa_opt3861602675778688675t_assn(aTP_Lamm_gi(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_278
tff(fact_10484_ATP_Olambda__279,axiom,
    ! [Uu: assn,Uua: $o,Uub: $o] : aa_o_assn(aa_o_fun_o_assn(aTP_Lamm_gh(Uu),(Uua)),(Uub)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,(Uub) = (Uua))) ).

% ATP.lambda_279
tff(fact_10485_ATP_Olambda__280,axiom,
    ! [Uu: assn,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_gk(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_280
tff(fact_10486_ATP_Olambda__281,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_em(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_281
tff(fact_10487_ATP_Olambda__282,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_el(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_282
tff(fact_10488_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_gc(Uu),Uua),Uub) = real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_283
tff(fact_10489_ATP_Olambda__284,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gb(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_284
tff(fact_10490_ATP_Olambda__285,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mm(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_285
tff(fact_10491_ATP_Olambda__286,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_gg(Uu),Uua),Uub) = aa_o_assn(pure_assn,Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),Uua)) ).

% ATP.lambda_286
tff(fact_10492_ATP_Olambda__287,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_gf(Uu),Uua),Uub) = aa_o_assn(pure_assn,Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),Uua)) ).

% ATP.lambda_287
tff(fact_10493_ATP_Olambda__288,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_ik(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_ij(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_288
tff(fact_10494_ATP_Olambda__289,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lr(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_289
tff(fact_10495_ATP_Olambda__290,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lp(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_290
tff(fact_10496_ATP_Olambda__291,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_lo(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_291
tff(fact_10497_ATP_Olambda__292,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ln(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_292
tff(fact_10498_ATP_Olambda__293,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ih(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_293
tff(fact_10499_ATP_Olambda__294,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ig(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_294
tff(fact_10500_ATP_Olambda__295,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_jq(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_295
tff(fact_10501_ATP_Olambda__296,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_jn(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_296
tff(fact_10502_ATP_Olambda__297,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(aa_set5977240410424667023T_assn(aTP_Lamm_ki(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uuc),Uub)),
        aa_o_assn(pure_assn,
          ( ( Uua = vEBT_set_vebt(Uuc) )
          & aa_nat_o(vEBT_invar_vebt(Uuc),Uu) ))) ).

% ATP.lambda_297
tff(fact_10503_ATP_Olambda__298,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: $o] : aa_o_assn(aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_di(Uu),Uua),Uub),(Uuc)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,(Uuc) = aa_nat_o(vEBT_vebt_member(Uu),Uub))) ).

% ATP.lambda_298
tff(fact_10504_ATP_Olambda__299,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_dr(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uuc = vEBT_vebt_succ(Uu,Uub))) ).

% ATP.lambda_299
tff(fact_10505_ATP_Olambda__300,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_dq(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uuc = vEBT_vebt_pred(Uu,Uub))) ).

% ATP.lambda_300
tff(fact_10506_ATP_Olambda__301,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ft(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_301
tff(fact_10507_ATP_Olambda__302,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fu(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_302
tff(fact_10508_ATP_Olambda__303,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_ij(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_303
tff(fact_10509_ATP_Olambda__304,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_io(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,Uuc = vEBT_vebt_mint(nth_VEBT_VEBT(Uua,Uu)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua,Uub)) ).

% ATP.lambda_304
tff(fact_10510_ATP_Olambda__305,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_in(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,Uuc = vEBT_vebt_maxt(nth_VEBT_VEBT(Uua,Uu)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua,Uub)) ).

% ATP.lambda_305
tff(fact_10511_ATP_Olambda__306,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_jy(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_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_306
tff(fact_10512_ATP_Olambda__307,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_js(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_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_307
tff(fact_10513_ATP_Olambda__308,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_jw(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_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_308
tff(fact_10514_ATP_Olambda__309,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ju(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_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_309
tff(fact_10515_ATP_Olambda__310,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: $o] : aa_o_assn(aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aa_set2974300032405964540o_assn(aTP_Lamm_dk(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Uu,Uua),Uub)),aa_o_assn(pure_assn,(Uud) = aa_set_nat_o(member_nat(Uuc),Uua))) ).

% ATP.lambda_310
tff(fact_10516_ATP_Olambda__311,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_lq(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(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lp(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_311
tff(fact_10517_ATP_Olambda__312,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa_nat_VEBT_VEBT(aa_nat6729221662743847087T_VEBT(aa_nat1426624462703894328T_VEBT(aa_VEB257809287009931607T_VEBT(aa_lis5232481317877780700T_VEBT(aTP_Lamm_ip(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uud)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uuc) ),
        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uud),Uue)),Uu,Uua,Uub),
        $ite(
          ( ( Uuc = Uud )
          & ( Uuc = Uue ) ),
          vEBT_Node(none_P5556105721700978146at_nat,Uu,Uua,Uub),
          $let(
            xn: nat,
            xn:= 
              $ite(Uuc = Uud,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(Uub))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(Uua,the_nat(vEBT_vebt_mint(Uub)))))),Uuc),
            $let(
              minn: nat,
              minn:= 
                $ite(Uuc = Uud,xn,Uud),
              $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,Uu),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(Uua)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(Uua,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                    $let(
                      newlist: list_VEBT_VEBT,
                      newlist:= list_u1324408373059187874T_VEBT(Uua,h,newnode),
                      $ite(
                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Uub,h),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                $ite(
                                  xn = Uue,
                                  $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,Uu),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)))))) ),
                                  Uue ))),Uu,newlist,sn) ),
                        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                              $ite(xn = Uue,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,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Uue))),Uu,newlist,Uub) ) ) ),
                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uud),Uue)),Uu,Uua,Uub) ) ) ) ) ) ) ).

% ATP.lambda_312
tff(fact_10518_ATP_Olambda__313,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa_nat_option_nat(aa_nat1489030058641307643on_nat(aa_nat1972073180414880072on_nat(aa_VEB5583160467804174207on_nat(aa_lis5680555565288705232on_nat(aTP_Lamm_iu(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uud),
        some_nat(Uud),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),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,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),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(Uua)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(Uua,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & 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,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(Uua,h),l)),
                  $let(
                    sc: option_nat,
                    sc:= vEBT_vebt_succ(Uub,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,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(Uua,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% ATP.lambda_313
tff(fact_10519_ATP_Olambda__314,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: list_VEBT_VEBT,Uuc: vEBT_VEBT,Uud: nat,Uue: product_prod_nat_nat] :
      aa_Pro3676457433925622191T_VEBT(aa_nat6988169923125987140T_VEBT(aa_VEB5764018861059009915T_VEBT(aa_lis2657662162680668108T_VEBT(aa_nat7935216168507688033T_VEBT(aTP_Lamm_iq(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),vEBT_Node(Uu,Uua,Uub,Uuc),aa_Pro3676457433925622191T_VEBT(produc3169358591047799142T_VEBT(aa_nat1426624462703894328T_VEBT(aa_VEB257809287009931607T_VEBT(aa_lis5232481317877780700T_VEBT(aTP_Lamm_ip(Uua),Uub),Uuc),Uud)),Uue)) ).

% ATP.lambda_314
tff(fact_10520_ATP_Olambda__315,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa_nat_VEBT_VEBT(aa_nat6729221662743847087T_VEBT(aa_nat1426624462703894328T_VEBT(aa_VEB257809287009931607T_VEBT(aa_lis5232481317877780700T_VEBT(aTP_Lamm_ir(Uu),Uua),Uub),Uuc),Uud),Uue) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uud),Uud,Uuc),
        $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,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(Uua))
            & ~ ( ( Uuc = Uud )
                | ( Uuc = Uue ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uud),Uuc,Uud)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Uue))),Uu,list_u1324408373059187874T_VEBT(Uua,h,vEBT_vebt_insert(nth_VEBT_VEBT(Uua,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(Uua,h)),vEBT_vebt_insert(Uub,h),Uub)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uud),Uue)),Uu,Uua,Uub) ) ) ) ).

% ATP.lambda_315
tff(fact_10521_ATP_Olambda__316,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ev(Uu),Uua) = Uu ).

% ATP.lambda_316
tff(fact_10522_ATP_Olambda__317,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ew,Uu),Uua) = Uu ).

% ATP.lambda_317
tff(fact_10523_ATP_Olambda__318,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_kp((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_318
tff(fact_10524_ATP_Olambda__319,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_hm,Uu) = Uu ).

% ATP.lambda_319
tff(fact_10525_ATP_Olambda__320,axiom,
    ! [Uu: assn] : aa_assn_assn(aTP_Lamm_cd,Uu) = Uu ).

% ATP.lambda_320
tff(fact_10526_ATP_Olambda__321,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ce,Uu) = Uu ).

% ATP.lambda_321
tff(fact_10527_ATP_Olambda__322,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cf,Uu) = Uu ).

% ATP.lambda_322
tff(fact_10528_ATP_Olambda__323,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cg,Uu) = Uu ).

% ATP.lambda_323
tff(fact_10529_ATP_Olambda__324,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ch,Uu) = Uu ).

% ATP.lambda_324
tff(fact_10530_ATP_Olambda__325,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_by,Uu) = zero_zero_complex ).

% ATP.lambda_325
tff(fact_10531_ATP_Olambda__326,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_bz,Uu) = zero_zero_real ).

% ATP.lambda_326
tff(fact_10532_ATP_Olambda__327,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ca,Uu) = zero_zero_rat ).

% ATP.lambda_327
tff(fact_10533_ATP_Olambda__328,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_en,Uu) = zero_zero_complex ).

% ATP.lambda_328
tff(fact_10534_ATP_Olambda__329,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_eo,Uu) = zero_zero_real ).

% ATP.lambda_329
tff(fact_10535_ATP_Olambda__330,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cb,Uu) = zero_zero_nat ).

% ATP.lambda_330
tff(fact_10536_ATP_Olambda__331,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_ep,Uu) = zero_zero_int ).

% ATP.lambda_331
tff(fact_10537_ATP_Olambda__332,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cc,Uu) = zero_zero_int ).

% ATP.lambda_332
tff(fact_10538_ATP_Olambda__333,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_am,Uu)
    <=> $false ) ).

% ATP.lambda_333
tff(fact_10539_ATP_Olambda__334,axiom,
    ! [Uu: complex] :
      ( aa_complex_o(aTP_Lamm_al,Uu)
    <=> $false ) ).

% ATP.lambda_334
tff(fact_10540_ATP_Olambda__335,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_an,Uu)
    <=> $false ) ).

% ATP.lambda_335
tff(fact_10541_ATP_Olambda__336,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_ao,Uu)
    <=> $false ) ).

% ATP.lambda_336
tff(fact_10542_ATP_Olambda__337,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_ap,Uu)
    <=> $false ) ).

% ATP.lambda_337
tff(fact_10543_ATP_Olambda__338,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_kn,Uu)
    <=> $true ) ).

% ATP.lambda_338

% Helper facts (17)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X8: $o,Y: $o] :
      ( ( (X8) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X8)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X8: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X8)),(Y))
      | ( (X8) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X8: int,Y: int] :
      ( ( X8 != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X8: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X8: nat,Y: nat] :
      ( ( X8 != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X8: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X8: num,Y: num] :
      ( ( X8 != Y )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X8: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X8: real,Y: real] :
      ( ( X8 != Y )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X8: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X9: real] : aa_real_o(P,X9) ) ) ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X8: complex,Y: complex] :
      ( ( X8 != Y )
      | aa_complex_o(fequal_complex(X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X8: complex,Y: complex] :
      ( ~ aa_complex_o(fequal_complex(X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X8: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( X8 != Y )
      | aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X8: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X8: product_prod_int_int,Y: product_prod_int_int] :
      ( ( X8 != Y )
      | aa_Pro4592068669816069595_int_o(fequal6010348034681699726nt_int(X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X8: product_prod_int_int,Y: product_prod_int_int] :
      ( ~ aa_Pro4592068669816069595_int_o(fequal6010348034681699726nt_int(X8),Y)
      | ( X8 = Y ) ) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    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))),n)) ).

tff(conj_1,conjecture,
    aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(n,s),ti),vEBT_vebt_inserti(ti,x),vEBT_Intf_vebt_assn(n,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,s),insert_nat(x,bot_bot_set_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))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,n))))))) ).

%------------------------------------------------------------------------------